Orthogonal matrix polynomials and differential, difference and q-difference matrix operators

Programa de Doctorado “Matem´aticas” PhD Dissertation ORTHOGONAL MATRIX POLYNOMIALS AND DIFFERENTIAL, DIFFERENCE AND q-DIFFERENCE MATRIX OPERATORS. Author Ana Mart´ınez de los R´ıos Supervisor Prof. Dr. Antonio Dur´an Guarde˜no Co-supervisor Dr. Renato ´ Alvarez-Nodarse May 22, 2014 Contents Introduction v 1 Preliminaries. 1 1.1 Hypergeometric functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.1 Hypergeometric functions . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.2 Basic hypergeometric functions. . . . . . . . . . . . . . . . . . . . . . 3 1.2 Matrix Orthogonal Polynomials. . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Matrix Operators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.3.1 Matrix differential operators . . . . . . . . . . . . . . . . . . . . . . . . 12 1.3.2 Matrix difference operators . . . . . . . . . . . . . . . . . . . . . . . . 16 1.3.3 Matrix q-difference operators . . . . . . . . . . . . . . . . . . . . . . . 19 2 Discrete orthogonal polynomials 23 2.1 A method to solve the symmetry equations. . . . . . . . . . . . . . . . . . . . 25 2.2 Examples. ....................................... 26 2.2.1 Example with dgr(F1)=0 ......................... 26 2.2.2 Example with dgr(F1)=1 and unbounded support. . . . . . . . . . . 28 2.2.3 Example with dgr(F1)=1 and finite support . . . . . . . . . . . . . 30 2.2.4 Example with dgr(F1)=2 ......................... 31 3 The convex cone. 37 3.1 Two methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.2 Examples. ....................................... 40 3.2.1 Examples constructed by using the first method . . . . . . . . . . . . 40 3.2.2 Examples constructed by using the second method . . . . . . . . . . 44 4 Matrix q-polynomials. 51 4.1 A family of q-polynomials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 4.2 Matrix q-hypergeometric function. . . . . . . . . . . . . . . . . . . . . . . . . 56 4.3 The 2 ×2case. .................................... 57 4.3.1 The little q-Jacobi basis. . . . . . . . . . . . . . . . . . . . . . . . . . . 58 4.3.2 The matrix hypergeometric function. . . . . . . . . . . . . . . . . . . 60 4.3.3 The TTRR and the Rodrigues’ formula . . . . . . . . . . . . . . . . . 60 i ii CONTENTS 5 Continuous matrix polynomials 67 5.1 The weight matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 5.2 Two differential operators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 5.3 Matrix hypergeometric function. . . . . . . . . . . . . . . . . . . . . . . . . . . 84 5.4 Three-term recurrence relation . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 5.5 Gegenbauer and Racah polynomials. . . . . . . . . . . . . . . . . . . . . . . . 88 ii Introduction In this thesis we present a series of result that are framed in the theory of Matrix Orthogonal Polynomials, a branch of the very celebrated subject of Orthogonal Polynomials. The theory of orthogonal polynomials dates back from the end of the 19th century and its birth used to be attributed to the works of T. J. Stieltjes (1856-1894) and P. L. Chebyshev (1821-1894). The first discovered the orthogonality relation of the denominators of certain continuous fractions and to him is also attributed a primitive version of Farvard’s theorem. He also introduced what is now known as the moments problem, given a sequence (µn)nfind a measure dµ such that µn=∫xndµ(x). Chebyshev, among other results, introduced the Chebyshev polynomials (nowadays known as Chebyshev polynomials of the first kind) Tnin order to find the best polynomial approximation for a continuous function. However several authors had already introduced particular examples that had came to them via physical or mathematical problems. For example, Adrien Marie Legendre (1752- 1833) introduced the Legendre polynomials (later generalized by Jacobi) related with the resolution of planetary motion problems. After him, Charles Hermite (1822-1901) defined the Hermite polynomial for the study of expansion series in R, although these polynomials had already been considered and studied by Laplace in the context of probability theory. In relation with the integral ∫∞ x e−x xdx, E. N. Laguerre (1834-1886) introduced the Laguerre polynomials, that were generalized by N. Sonin. In 1859 a work by K. G. J. Jacobi (1804-1851) gave a generalization of the Legendre polynomials, known as Jacobi polynomials. The first monograph dedicated to the subject of orthogonal polynomials appeared in 1939, Orthogonal polynomials by Gabor Szeg¨o [97]. All these families of orthogonal polynomials happened to have some interesting properties in common, in particular they are all eigenfunctions of a second order differential operator of the form (1) D(⋅)=σ(x)d2 dx2+τ(x)d dx, where σis a polynomial of degree at most two and τis a polynomial of degree exactly one. This property characterizes the families of Jacobi, Hermite and Laguerre polynomials, as it was proven by S. Bochner in 1929, [8]. Some other properties were shown to characterize these families. In 1887 Sonin proved that the only families of orthogonal polynomials whose derivatives are also orthogonal were the Jacobi, Hermite and Laguerre polynomials. v vi INTRODUCTION We also want to mention a third characterization proposed by F. Tricomi, that is the fact that these families can be expressed in terms of a Rodrigues formula. pn(x)=bn ρ(x)dn dxn(σn(x)ρ(x)), where ρ(x)is the weight with respect to which the sequence is orthogonal. A very handy characterizing property of these families of orthogonal polynomials is a distributional one, the weights associated to these families satisfy a Pearson equation (σ(x)ρ(x))′=τ(x)ρ(x). These families are called classical orthogonal polynomials and their associated weights classical weights. The classical orthogonal polynomials constitute a very important class of families of orthogonal polynomials because of their properties and applications. They can be expressed in terms of hypergeometric functions 2F1(−n, a b∣x)=n ∑ k=0(−n)k(a)k (b)k xk k!. In 1949 two authors published what we could see as the seminal works for this thesis. The first one, W. Hahn (1911-1998) extended with ¨ Uber Orthogonalpolynome die q-Differenzengleichungen gen¨ugen, [73], the classical orthogonal polynomials by generalizing their properties. For qand wfixed numbers with q≠1 let L be the linear operator Lf(x)=f(qx +w)−f(x) (q−1)x+w. Hahn proposed the following problem1: determine all sequences of orthogonal polynomials (pn)nsuch that 1. (Lpn(x))nis also a sequence of orthogonal polynomials. 2. For all n≥0, pn(x)satisfies an operator equation of the form ˜σ(x)L2pn(x)+˜τ(x)Lpn(x)=λnpn(x)˜σ(x),˜τ(x)∈C[x], λn∈C. 3. pn(x)=Ln(R0(x)R1(x)⋯Rn(x)ρ(x))ρ(x)−1, where R0(x)is a polynomial and Ri(x)=Ri+1(qx +w)and ρ(x)is the weight with respect to which (pn)nis orthogonal. In this article Hahn proved that these properties determine one and the same class of orthogonal polynomials and these polynomials can be constructed in terms of basic hypergeometric functions. For the particular case q=1, w=1 we recover what it is known as the classical discrete orthogonal polynomials. The difference equation satisfied by these polynomials, ¯σ(x)∆∇p(x)+¯τ(x)∆p(x)=λp(x), 1Hahn’s problem involves two more equivalent properties that we have skipped because we do not consider them in the matrix case. vi Introduction vii where ∆(p)(x)=p(x+1)−p(x),∇(p)(x)=p(x)−p(x−1), is a discretization in the lattice x(s)=sof the differential equation satisfied by the classical orthogonal polynomials. This particular problem was considered separately by O. E. Lancaster [83], and P. Lesky [86]. They showed that the only families of classical discrete orthogonal polynomials are the Charlier, Meixner, Krawchuck and Hahn polynomials. These families satisfy a discrete orthogonality property κ ∑ k=0 pn(x)pm(x)ρ(x)=hnδn,m, κ ∈Nor κ=∞. As the classical orthogonal polynomials they can be expressed in terms of hypergeometric functions and their orthogonalizing weights satisfy a Pearson equation ∆(¯σ(x)ρ(x))=¯τ(x)ρ(x). By setting q∈(0,1)and w=0 in Hahn’s problem we recover one of the first families of q-classical orthogonal polynomials discovered, the big q-Jacobi polynomials. The qclassical2orthogonal polynomials are families of orthogonal polynomials that satisfy a second order q-differential equation, ˜σ(qx)DqDq−1p(qx)+˜τ(qx)Dqp(qx)=λp(qx), where Dqf(x)=f(qx)−f(x) (q−1)xis the q-differential operator. More characterizing properties of the q-classical orthogonal polynomials were accomplished at the end of the 20th century and the beginning of the 21st, as well as a complete classification of them (see [3] and [90]). In particular the q-classical polynomials were shown to satisfy a Rodrigues equation and their associated weights a Pearson equation. These orthogonal polynomials can be described in terms of basic hypergeometric series rϕs[a1,⋯, ar b1,⋯, bs ;q, x]=∞ ∑ k=0(a1,⋯, ar;q)k (b1,⋯, bs;q)k xk (q;q)k . According to the representation of the classical and the classical discrete orthogonal polynomials in terms of hypergeometric functions the Askey-scheme provides a hierarchical classification of them, establishing also some limit relations among the different families of orthogonal polynomials (see for instance [4]). The families of classical orthogonal polynomials can be obtained as limiting cases of certain q-polynomials written in terms of 4ϕ3, the Askey Wilson polynomials, what entailed the appearance of the q-Askey-scheme, where polynomials expressed in terms of basic hypergeometric functions are treated and limit relations among them are shown (see [77]). A different treatment of the classical discrete and q-classical orthogonal polynomials is due to A. F. Nikiforov and V. B. Uvarov and is based on the analysis of the discretization 2The term q-classical makes reference to a bigger class of orthogonal polynomials, those that satisfy an operator equation that results form discretizing the second order differential equation in the q-quadratic lattice. However in the present work we accomplished the term q-classical to this subclass, that used to be referred as Hahn class. vii viii INTRODUCTION of the second order differential equations satisfied by the classical orthogonal polynomials in the uniform lattice x(s)=s, and the non-uniform lattice x(s)=c1qs+c2q−s+c3, [93]. The other author whose work is an starting point for this memory is M. G. Krein (1907-1989). His publications from 1949, The fundamental propositions of the theory of representations of Hermitian operators with deficiency index (m, m), [82], and Infinite J- matrices and a matrix-moment problem, [81], are considered as the beginning of the theory of matrix orthogonal polynomials. In this theory one deals with the sesquilinear form defined by a matrix of measures W in the space of matrix polynomials: (2) ⟨P, Q⟩W=∫PdWQ∗. Under mild assuptions on the matrix of measures W, one can always construct sequences of orthogonal matrix polynomials with respect to W. These matrices of measures will be called weight matrices. Since its birth and until the last decade of the previous century, the works concerning the theory of matrix orthogonal polynomials were somehow sporadic (see for instance [10], [65]). This situation has radically changed in the last two decades, and since the end of the 20th century more systematic works have appeared, bringing a solid and structured knowledge concerning this theory. Many basic results from the scalar theory of orthogonal polynomials have been extended to the matrix theory. Among them, there have been discovered recurrence formulas, with extension of Favard’s theorem, algebraic aspects, properties of the zeros and Gaussian quadrature formulas (see e.g [22], [25], [26], [27], [47], [54], [58], [96]). Also some density problems and problems regarding matrix moments have been studied (e. g. [48], [49], [50], [53], [84], [85]). The study of asymptotic properties for the matrix orthogonality has also been considered (see e.g. [20], [21], [29], [55], [87], [88]). It is worth mentioning that these properties haven’t been obtained by using tools derived from Riemmann-Hilbert, but from quadrature formulas. However taking into account the efficiency of such methods for the scalar case it would be convenient to develop the basis of these tools in the matrix setting that allow its use in the matrix case. The construction and study of examples of matrix orthogonal polynomials satisfying second order differential equations (e.g. [9], [12], [13], [14], [15], [31], [30], [36], [37], [38], [39], [40], [41], [42], [43], [44], [45], [66], [70], [71], [95], [72], [94]), have shown that these families also posses the wealth of structural properties, as their scalar relatives (e.g. [32], [33], [34], [39], [52]). The theory of matrix orthogonal polynomials is linked to different areas of the mathematics, pointing out the power of this theory. For instance, in [58] the authors established the link between matrix orthogonal polynomials and scalar polynomials satisfying 2N+1- recurrence relations. Other works, as e.g. [68], show the connection with probability theory, and random matrices [74]. More applications of the theory of matrix orthogonal polynomials can be found in e.g. [96]. For a more complete view of the matrix orthogonal polynomials theory we reefer the readers to [19], [51] and references therein. The existence of singular matrices as well as the non-commutativity of the matrix product difficult the treatment of these objects and force us to developed new ideas and techniques for its study. viii Introduction xv with respect to a weight matrix Wof the form (1.18) (see Theorem 1.3.21). For the particular case of a second order q-difference operator D(P)(qx)=P(qx+1)F1(qx)+P(qx)F0(qx)+P(qx−1)F−1(qx), Fk(x)∈CN×N[x−1], the symmetry equations are given by F0(qx)W(qx)=W(qx)F0(qx)∗, x ∈N, F1(qx−1)W(qx−1)=qW (qx)F−1(qx)∗, x ∈N/{0}, and the boundary condition W(1)F−1(1)∗=0. q2xF1(qx)W(qx)→0,as x→∞, qx(F1(qx)W(qx)−W(qx)F1(qx)∗)→0,as x→∞. We provide a method to construct second order q-difference operators and q-weights that satisfy the symmetry conditions. This method is based on that developed for the discrete case and profits of the decomposition of the q-weight matrix W(qx)=T(qx)T(qx)∗. By making use of this method we construct an example of arbitrary size that constitute, as long as we know, the first non trivial example of matrix q-polynomials satisfying qdifference equations. As an extension of the matrix hypergeometric functions introduced by Tirao [98] we define a matrix basic hypergeometric function. Such a matrix function can be used to describe the rows of the matrix orthogonal polynomials which are eigenfunctions of a particular second order q-difference operator, whose coefficients are related with the parameters of the matrix basic hypergeometric function. In Chapter 4we also carry an extensive study of the particular case of size N=2, exhibited below. Let aand bbe two real numbers such that 0 <a<q−1and b<q−1. Consider the matrices M,Land Ugiven by U=(0v(1−q)(q−1−a) 0 0 ), M =(1v 0 0), L =elog(q)M=(q−v(1−q) 0 1 ). The monic orthogonal polynomials (Pn)nwith respect to W(qx)=ax(bq;q)x (q;q)x Lx(L∗)x, satisfy then the matrix q-difference equation Pn(qx−1)(q−x−1)L−1+Pn(qx)(U−q−x(L−1+aL))+Pn(qx+1)(aq−x−abq)L=ΛnPn(qx), where Λn=(−q−n−1−abqn+2v(1−q)(abqn+1−q1−n+q−1−a) 0−q−n−abqn+1). By profiting the factorization of the weight matrix, we provide formulas for the matrix polynomials Pnin terms of little q-Jacobi polynomials. We also make use of the matrix xv xvi INTRODUCTION basic hypergeometric function introduced in Section 4.2 to give an alternative expression of the polynomials, and by making use of this expression we find a three term recurrence relation for the 2 ×2 polynomials. Also a Rodrigues formula is found. The content of this chapter can be found in [1]. Chapter 5is related to the already mentioned connection between representation theory and matrix orthogonal polynomials is showing to be a very fruitful one. In [78], and [79] the authors studied the full spherical functions associated to the particular case (SU(2)×SU(2), SU(2))and related them with a family matrix orthogonal polynomials. By profiting of the machinery of the representation theory the authors of [78] and [79] showed that these matrix orthogonal polynomials satisfy a second order differential equation, and also a first order differential equation. In Chapter 5we introduce a family of weight matrices labeled by a positive number (W(ν))ν>0, and their sequences of orthogonal polynomials (P(ν) n)n. For the particular case of ν=1 we recover the example in [78]. We define the weight matrices via its LDU-decomposition and we prove another equivalent and useful expression for the weights in terms of Gegenbauer polynomials. These weight matrices can be reduced to lower size, explicitly W(ν)(x)=Yt N 2⎛⎝W(ν) +(x)0 0W(ν) −(x)⎞⎠YN 2 , where YN 2 are the matrices given by Yp+1 2=1 2√2(Ip+1Jp+1 −Jp+1Ip+1), Yp=1 2√2⎛⎜⎝ Ip0Jp 0√2 0 −Jp0Ip⎞⎟⎠, p ∈N, where Ipdenotes the identity matrix of size pand Jp=⎛⎜⎜⎜⎜⎜⎜⎝ 0 0 ⋯0 1 0 0 ⋯1 0 ⋰ 0 1 ⋯0 0 1 0 ⋯0 0. ⎞⎟⎟⎟⎟⎟⎟⎠ It is also worth to notice that W(ν) +(x)∈Cr1×r1and W(ν) −(x)∈Cr2×r2, where r1=⌈N 2⌉, r2=⌊N 2⌋, and ⌈⌉and ⌊⌋are the ceiling and floor functions, ⌈x⌉=min{m∈Z∣m≥x},⌊x⌋max{m∈Z∣m≤x}. Moreover they admit no more reduction. For each ν>0 we construct a pair of differential operators, one of order two D(ν)and the other of order one E(ν), that satisfy the symmetry equations and boundary conditions for W(ν). The operator D(ν)can also be reduced to lower size via the composition with the matrices Y`, but the operator E(ν)does not reduce to lower size. These operators can be combined to obtain a third differential operator, D(ν) Φ,Ψ=d2 dx2Φ(ν)+d dxΨ(ν). xvi Introduction xvii The existence of such operator implies the Hermiticity of Φ(x)(ν)W(ν)(x)and that the weight matrices W(ν)satisfy a Pearson-type equation as in [12], (Φ(x)(ν)W(ν)(x))′=Ψ(ν)(x)W(ν)(x). By exploiting the expression of the weights in terms of scalar Gegenbauer polynomials, we can prove that in fact W(ν+1)(x)=Φ(x)νWν(x), connecting the orthogonal polynomials (P(ν) n)nwith (P(ν+1) n)n. Moreover we can prove that the basic differential operator d dx is a forward shift operator, i.e. if (P(ν) n)nis the sequence of monic orthogonal polynomials with respect to W(ν), d dx (P(ν) n)=nP(ν+1) n−1. And by carrying an integration by parts on ⟨d dx P, Q⟩(ν)we can also give an operator, (T(ν)Q)(x)=dQ dx (x)(Φ(ν)(x))∗+Q(x)(Ψ(ν)(x))∗, that satisfies T(ν)P(ν+1) n−1=K(ν) nP(ν) n, for certain matrices K(ν) nthat can be determined from Φ(ν)(x)and Ψ(ν)(x), that is T(ν)is a backward shift operator. By combining the forward shift and backward shift operators we can get a compact Rodrigues formula for the sequences of orthogonal polynomials (P(ν) n)n, P(ν) n(x)=G(ν) n dn dxn(W(ν+n)(x))W(ν)(x)−1, where G(ν) nare diagonal matrices. We also provide expressions of the matrix orthogonal polynomials in terms of matrixvalued hypergeometric functions as in [98]. From this expression we can calculate the three term recurrence relation satisfied by the polynomials. From the LDU decomposition of the weight matrix (5.5), and by performing an appropriate change of variables, we find the expression of the matrix orthogonal polynomials (P(ν) n)nin terms of Gegnbauer and Racah polynomials. We end this introduction by displaying an explicit example of size N=3 of the weight matrices W(ν)and the differential operators D(ν)and E(ν). For ν>0 we define the weight matrix W(ν)in (−1,1)by W(ν)(x)=(1−x2)(ν−1/2)(2+ν)⎛⎜⎜⎝ 1x2x2(ν+1)−1 2ν+1 xx2ν+ν+1 2ν+1x 2x2(ν+1)−1 2ν+1x1⎞⎟⎟⎠. The weight matrix W(ν)can be decomposed in a two block matrix: W(ν)(x)=(1−x2)(ν−1/2)(2+ν)Y3 2⎛⎜⎜⎝ x2(ν+1)+ν 2ν+1√2x0 √2xνx2+ν+1 2ν+10 0 0 −2(x−1)(x+1)(1+ν) 2ν+1 ⎞⎟⎟⎠Y+ 3 2 , xvii xviii INTRODUCTION where Y3 2=1 √2⎛⎜⎝ 1 0 1 0√2 0 −1 0 1⎞⎟⎠. The sequence of monic orthogonal polynomials (P(ν) n)nwith respect to W(ν)satisfy the following differential equations: (1−x2)(P(ν) n(x))′′ +(2ν+3)(P(ν) n(x))′⎛⎜⎝−x(2ν+3)2 0 1−x(2ν+3)0 0 2 −x(2ν+3)⎞⎟⎠ (7) +P(ν) n(x)⎛⎜⎝−(v−1)(3+ν)0 0 0 1 −(ν−1)(3+ν)0 0 0 −(ν−1)(3+ν)⎞⎟⎠=Λn(D(ν))P(ν) n(x), (8) (P(ν) n(x))′⎛⎜⎝−x1 0 −1/2 0 1/2 0−1x⎞⎟⎠+P(ν) n(x)⎛⎜⎝−3−ν0 0 0−2 0 0 0 ν−1⎞⎟⎠=Λn(E(ν))P(ν) n(x) (9) where Λn(D(ν))=⎛⎜⎝(2ν+3)−1−(ν−1)(3+ν)0 0 0(2ν+3)−(ν−1)(3+ν)0 0 0 (2ν+3)−1−(ν−1)(3+ν)⎞⎟⎠, Λn(E(ν))=⎛⎜⎝−4−ν0 0 0−2 0 0 0 ν⎞⎟⎠. Moreover the sequences of monic orthogonal polynomials (P(ν) n)satisfy dk dxkP(ν) n(x)=(n−k+1)kP(ν+k) n−k(x),for all k≤n. The content of this chapter can be found in [80]. xviii Chapter 1 Preliminaries In this chapter we introduce some results concerning the theory of matrix orthogonal polynomials that will be used in the forward chapters. We also provide some useful formulas and establish some notation that will be used in the rest of the thesis. 1.1 Hypergeometric and basic hypergeometric function. Far away from being a deep treatment in hypergeometric and basic hypergeometric functions, we establish here some notation and define some important functions that will be used in this thesis. We also list a few interesting and useful summation formulas that will be needed in the forthcoming chapters. For a deeper understanding and study we refer the reader to [6], [63], [76], [77], among others. 1.1.1 Hypergeometric functions For a∈C,k∈Nwe set (a)kfor the ascending Pochhammer symbol, that is (a)k=a(a+1)(a+2),⋯,(a+k−1)=Γ(a+k) Γ(a). We can extend this function for negative integers by the convention (a)−k=⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ 0 if k≥a+1 Γ(a+1) Γ(a−k+1)if k<a+1 For a, b, c ∈Cthe Gauss hypergeometric function is defined by 1+ab cx+a(a+1)b(b+1) c(c+1)x2 2! +a(a+1(a+2)b(b+1)(b+2) c(c+1)(c+2)x3 3! +⋯=∞ ∑ k=0(a)k(b)k (c)k xk k!, whenever it makes sense. Notice that if c∈Zwith c<0 then, this function is not defined unless a∈Zwith a<0 and c≤a(or the same but with b). The Gauss hypergeometric function converges for ∣x∣<1 and for x=1 when R(c−a−b+1)>0. It satisfies the second order differential equation (1.1) x(1−x)d2 dx2(f(x))+(c−(a+b+1)x)d dx(f(x))−abf(x)=0. 1 2CHAPTER 1. PRELIMINARIES. We will denote the Gauss hypergeometric function by 2F1(a, b;c∣x)=2F1(a, b c∣x)=∞ ∑ k=0(a)k(b)k (c)k xk k!. If n∈Nthen, 2F1(−n,b c∣x)is a polynomials in xof degree nthat satisfies the second order differential equation (1.1). For a1, a2,⋯, ar∈Cand b1, b2,⋯, bp∈Cthe generalized hypergeometric function is given by (1.2) rFp(a1, a2,⋯, ar b1, b2,⋯, bp∣x)=∞ ∑ k=0(a1)k(a2)k⋯(ar)k (b1)k(b2)k⋯(bp)k xk k!. We now define balanced,well-poissed and very-well-poised series. Definition 1.1.1. A hypergeometric series p+1Fp(a1, a2,⋯,ap+1 b1, b2,⋯,bp∣x)is called balanced (or Saalsch¨utzian) if it satisfies 1+p+1 ∑ k=0 ak=p ∑ k=0 bk. We say that p+1Fpis well-poised if it satisfies 1+a1=b1+a2=b2+a3=⋯=bp+ap+1. And we say that p+1Fpis a very-well-poised series if it is well-poised and a2=a1 2+1. Summation formulas We list here some summation formulas for hypergeometric series. They can be found in e.g. [5], [6]. •Gauss summation formula. For R(c−a−b)>0 (1.3) 2F1(a, b c∣x)=Γ(c)Γ(c−a−b) Γ(c−a)Γ(c−b). •Kummer’s summation formula (1.4) 2F1(a, b 1+a−b∣−1)=Γ(1+a−b)Γ(1+1 2a) Γ(1+a)Γ(1+1 2a−b). •Pfaff-Saalsch¨utz formula. For a 3F2balanced series we have (1.5) 3F2(−n, a, b c, a +b−n−c+1∣1)=(c−a)n(c−b)n (c)n(c−a−b)n . •Whipple’s transformations. For a 4F3balanced series we have 4F3(−n, a, b, c d, e, f ∣1)=(e−a)n(f−a)n (e)n(f)n 4F3(−n, a, d −b, d −c d, a +1−n−e, a +1−n−f∣1).(1.6) •Dougall summation formula. 5F4(a, a/2+1, c, d, −m a/2, a −c+1, a −d+1, a +m+1∣1)=(a+1)m(a−c−d+1)m (a−c+1)m(a−d+1)m .(1.7) 1.1. Hypergeometric functions. 3 1.1.2 Basic hypergeometric functions. In forthcoming chapters and sections we will occasionally work with q-orthogonal polynomials, for what we need to introduce some notation and definitions. For the rest of the thesis qstands for a real number with 0 <q<1. For m∈Nthe q-shifted factorial, (or simply q-factorial or q-Pochhammer symbol), is defined by (1.8) (α;q)m=(1−α)(1−qα)(1−q2α)⋯(1−qm−1α),(α;q)∞=∞ ∏ k=0(1−aqk), and the multiple q-shifted factorials (α1, α2,⋯, αp;q)m=p ∏ k=1(αk;q)m. We can extend the definition of q-shifted factorial for m∈Zby using the formula (1.9) (αq;q)m=(α;q)∞ (αqm;q)∞ , valid for a negative integer mprovided αqm≠q−nfor any n∈N. The q-binomial coefficient is [n k]q=(q;q)n (q;q)k(q;q)n−k . We define the basic hypergeometric serie of parameters α1, α2,⋯, αp∈Cand β1, β2,⋯, βr∈ Cby pφr[α1, α2,⋯, αp β1, β2,⋯, βq ;q, x]=∞ ∑ k=0(α1, α2,⋯, αp;q)k (β1, β2,⋯, βr;q)k xk (q;q)k , see [63]. If for some k=1,⋯, r,βk=q−nfor some n∈N, then the function pϕris not defined unless αj=q−mwith m<nfor some j=1,⋯, p. Notice that if αj=q−mfor some j=1,⋯, r then, pφr[α1, α2,⋯, αp β1, β2,⋯, βq ;q, x] is a polynomial in xof degree at most m. There is a q-analogue of the gamma function. The q-gamma function is defined by Γq(x)=(q;q)∞ (1−q)x−1(qx;q)∞ . It satisfies the functional equation Γq(x+1)=1−qx 1−qΓq(x). 4CHAPTER 1. PRELIMINARIES. There is also a notion of q-derivation and q-integration. We define the q-derivative of a function with derivative at x=0 by (1.10) Dqf(x)=⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ f(x)−f(qx) (1−q)xif x≠0 f′(0)if x=0. The q-integral or Jackson integral is the inverse of the q-derivative, (1.11) ∫a 0f(x)dqx=∞ ∑ k=0(aqk−aqk+1)f(aqk),∫b af(x)dqx=∫b 0f(x)dqx−∫a 0f(x)dqx. 1.2 Matrix orthogonal polynomials Let us denote by CN1×N2the vector space of matrices of size N1×N2whose entries are complex numbers, and write CNfor the vector space formed by the complex vectors of size N, and we will specify whenever it is needed whether we are working with row or column vectors. Definition 1.2.1. A matrix polynomial P(x)is a polynomial whose coefficients are matrices, all of them of the same size. We denote by CN1×N2[x]the vector space of all matrix polynomials of size N1×N2, CN1×N2[x]={m ∑ k=0 Mkxk∣Mk∈CN1×N2, m ∈N}. For k∈Nwe denote by CN1×N2 k[x]⊂CN1×N2[x]the set of all matrix polynomials of size N1×N2with degree at most k. In this work we focus on the particular case of square matrix polynomials, CN×N[x] and we will restrict the variable xto the real line. There are also very interesting works on the theory of matrix orthogonal polynomials on the complex plane, in particular on the unit circle (see e.g. [23], [64], [89]), but it is out of the scope of this thesis. The identity matrix in CN×Nis represented by I. The symbol Ei,j stands for the N×N matrix with entry (i, j)equal to 1 and 0 otherwise. It is not difficult to see that these matrices satisfy the following property (1.12) Ei,jEk,l =Ei,lδj,k. With this notation a matrix M∈CN×Ncan be written as M=N ∑ i,j=1 mijEij, where mij ∈C. For a matrix M∈CN×N, we denote by M∗its Hermitian conjugate, that is if M= (mi,j)N i,j=1then, M∗=(mj,i)N i,j=1. The spectrum of a matrix Mis denoted by σ(M), σ(M)={λi∈C∣det(M−λiI)=0}, and we will denote by ρ(M)its spectral radius, that is (1.13) ρ(M)=max λ∈σ(λ). The orthogonality will be defined with respect to a weight matrix. 1.3. Matrix Operators. 11 Definition 1.3.2. Let D∶CN×N[x]→CN×N[x]be a matrix operator taking matrix polynomials into matrix polynomails. We say that Dis symmetric with respect to a weight matrix Wif it satisfies (1.31) ⟨D(P), Q⟩=∫D(P)dWQ∗=∫P dWD(Q)∗=⟨P, D(Q)⟩. The following lemma establishes the relation between symmetric operators and matrix operators having matrix orthogonal polynomials as eigenfunctions. Lemma 1.3.3. [28] Let Wbe a weight matrix and (Pn)na sequence of orthonormal polynomials with respect to W. Let Dbe a matrix operator satisfying that for all polynomial P,D(P)is a polynomial with degree at most the degree of P. Then, Dis symmetric with respect to Wif and only if D(Pn)=ΛnPnfor certain sequence (Λn)nof Hermitian matrices. In this thesis we deal with three types of matrix operators 1. Matrix differential operators (1.32) D(P(x))=k0 ∑ k=0 dk dxk(P(x))Fk(x), Fk(x)∈CN×N[x]. 2. Matrix difference operators: (1.33) D(P(x))=r ∑ k=s sk(P(x))Fk(x),sk(P(x))=P(x+k), Fk(x)∈CN×N[x]. 3. Matrix q-difference operators, for 0 <q<1: (1.34) D(P(x))=r ∑ k=s Ek(P(x))Fk(x),Ek(P(x))=P(qkx), Fk(x)∈CN×N[x−1]. Notice that all these three types of matrix operators have matrix coefficients multiplying on the right, whereas the eigenvalues appearing in Lemma (1.3.3) are multiplying on the left. There are several reasons why we focus on these kinds of matrix operators (see [28], [38]). Notice that if (Pn)nis a sequence of orthogonal polynomials then, for any sequence of non-singular matrices (Λn)n, the sequence (ΛnPn)nis also a sequence of orthogonal polynomials, but in general (PnΛn)nis not a sequence of orthogonal polynomials. This is the reason why we multiply the eigenvalues on the left. The reason for the order of multiplication of the coefficients in the case of continuous weights and second order differential operators is explained in [28, Theorem 3.2 ], a result that can be easily generalized for the case of difference and q-difference operators: Theorem 1.3.4. The following conditions are equivalent: (i) Wis a continuous matrix weight (resp. discrete weight, q-weight) with a left-hand side second order differential operator (resp. matrix difference, q-difference operator) 12 CHAPTER 1. PRELIMINARIES. (ii) A non-singular matrix Sexists for which W=S∗WdSwhere Wd=ρI, with ρa classical scalar weight (resp. classical discrete weight, q-classical weight). In the following sections we investigate those matrix operators of the form (1.32), (1.33), (1.34) not raising the degree of matrix polynomials and being symmetric with respect to some weight. 1.3.1 Matrix differential operators In this section we study matrix-valued differential operators having a sequence of orthogonal polynomials as eigenfunctions. Such differential operators are related with a discrete-continuous non-commutative bispectral problem (for more information on bispectral problem we refer to [24], and to [69] for the non-commutative version). Recall that the orthogonality of a sequence is equivalent to a second order difference operator given by the three term recurrence relation (1.21). Then, if (Pn)nis a sequence of orthogonal polynomials being eigenfunctions of a differential operator D, then they are common eigenfunctions of two operators D(Pn)=k0 ∑ k=0 dk dxk(Pn(x))Fk(x)=ΛnPn(x), ΘPn(x)=AnPn+1(x)+BnPn(x)+CnPn−1(x)=xPn(x). If Dis a differential operator with a sequence of orthogonal polynomials as eigenfunctions, then Dis of the form [72] D(⋅)=k0 ∑ k=0 dk dxk(⋅)Fk(x), Fk(x)∈CN×N k[x]. The non-commutativity of the product of matrices entail the existence of sequences of matrix orthogonal polynomials being eigenfunctions of several independent matrix differential operator as one can see, for instance, in [14], what brings into the picture the concept of algebra of operators associated to a weight matrix. Let Wbe a weight matrix and (Pn)na sequence of orthogonal polynomials with respect to it. We define the set D(W)as (1.35) D(W)={D=k0 ∑ k=0 dk dxk(⋅)Fk(x)∣D(Pn)=ΛnPn, Fk∈CN×N k[x], k0∈N}. Notice that if (Qn)nis another sequence of orthogonal polynomials with respect to W then, there exists a sequence of non singular matrices (Mn)nsuch that Qn=MnPnand if D∈D(W)we have D(Qn)=D(MnPn)=MnD(Pn)=MnΛnM−1 nQn, so the definition of the algebra D(W)does not depend on the sequence chosen, but only on the weight matrix. It is worth observing that D(W)is a subalgebra of the Weyl algebra D(W)={D=k0 ∑ k=0 dk dxk(⋅)Fk(x)∣Fk∈CN×N[x], k0∈N}. 1.3. Matrix Operators. 13 Given a weight matrix W, and a sequence of orthogonal polynomials (Pn)nwith respect to W, a differential operator D∈D(W)is determined by the sequence of eigenvalues (Λn)n. Proposition 1.3.5. [72] Given a sequence (Pn)nof orthogonal polynomials with respect to W, let us consider the algebra D(W)defined in (1.35). Let Λ(D, n)=Λn(D)be the eigenvalue associated to Qn. Then, D↦Λ(D, n)is a representation of D(W)into CN×N. Moreover the sequence (Λn)nseparates elements of D(W). For D∈D(W),D=k0 ∑ k=0 dk dxkFk(x), with Fk=k ∑ j=0 Fj kxj. The eigenvalues Λnassociated to the monic orthogonal polynomials (Pn)nform a sequence of polynomials in nof degree at most the order of D, given by (1.36) Λn=k0 ∑ k=0(n k)k!Fk k. One can define a ∗-operation in the algebra of differential operators associated to a weight matrix via the adjoint of operators. Theorem 1.3.6. [72] For any D∈D(W)there exists a unique differential operator D∗∈ D(W)such that ⟨D(P), Q⟩=⟨P, D∗(Q)⟩, for all P, Q ∈CN×N[x]. We shall refer to D∗ as the adjoint of D. The map D↦D∗is a ∗-operation, and the orders of Dand D∗ coincide. By Lemma 1.3.3 it is clear that symmetric differential operators not raising the degree of polynomials are always in D(W). If we denote by S(W)⊂D(W)the subset of symmetric differential operators, we have that S(W)is a real form of the space D(W)(see [72]), D(W)=S(W)⊕iS(W). In the rest of the section we focus on second order matrix differential operators (1.37) D(⋅)=d2 dx2(⋅)F2(x)+d dx(⋅)F1(x)+(⋅)F0 The following theorem establishes criteria to assure the symmetry of a differential operator Dof degree 2 with respect to a weight matrix having a differentiable density with respect to the Lebesgue measure dW =W(x)dx. Theorem 1.3.7. [38] Let Dbe a second order matrix differential operator of the form (1.37) and Wa weight matrix with a differentiable density with respect to the Lebesgue measure dW =W(x)dx and supported on an interval I=(a, b)of the real line I⊆R(aand bfinite or infinite). If for all x∈I, the coefficients of Dand W(x)satisfy the symmetry equations F2(x)W(x)=W(x)F2(x)∗ 2(F2(x)W(x))′=F1(x)W(x)+W(x)F1(x)∗ (F2(x)W(x))′′ −(F1(x)W(x))′+F0W(x)=W(x)F∗ 0, 14 CHAPTER 1. PRELIMINARIES. and the boundary conditions lim x→{a,b}xnF2(x)W(x)=0,lim x→{a,b}xn((F2(x)W(x))′−F1(x)W(x))=0,for all n∈N, then, Dis symmetric with respect to W. The symmetry of a differential operator D, with respect to a weight matrix W, is also equivalent to a set of identities involving the moments of Wand the coefficients of D. Theorem 1.3.8. [38] Let Dbe a second order matrix differential operator of the form (1.37) and Wa weight matrix. Let us denote by µn=∫xndW (x)the moments associated to W, and we write Fk(x)=k ∑ i=0 Fi kxi, Fi k∈CN×N, k =0,1,2. for the coefficients of the differential operator (1.37). Then, the following properties are equivalent, (i) Dis symmetric with respect to W. (ii) For n=2,3,⋯, F2 2µn+F1 2µn−1+F0 2µn−2=µn(F2 2)∗+µn−1(F1 2)∗+µn−2(F0 2)∗. For n=1,2,⋯, 2(1−n)(F2 2µn+F1 2µn−1+F0 2µn−2)=F1 1µn+F0 1µn−1+µn(F1 1)∗+µn−1(F0 1)∗. And for n=0,1,⋯, n(1−n)(F2 2µn+F1 2µn−1+F0 2µn−2)−n(F1 1µn+F0 1µn−1)+F0 0µn=µn(F0 0)∗. As we have already mentioned, in the matrix case we lose some of the equivalences among those properties that characterize the classical orthogonal polynomials. In general, for a family of matrix orthogonal polynomials (Pn)nthat are eigenfunctions of a second order differential operator like (1.37), it is not true that the sequence formed by its derivatives, (P′ n)n, is a sequence of orthogonal polynomials with respect to some weight matrix. In [12] the authors established criteria to assure the orthogonality of (P′ n)n. Theorem 1.3.9. [12] Let Wbe a weight matrix supported in the interval I=(a, b)⊆R, and (Pn)nthe sequence of monic orthogonal polynomials with respect to W. Then, the following are equivalent, (i) There exists two matrix polynomials Φ,Ψwith dgr(Φ)≤2,dgr(Ψ)≤1satisfying det(Φ(x)W(x))≠0in Iand such that Φ(x)W(x)=W(x)Φ(x)∗, d dx (Φ(x)W(x))=Ψ(x)W(x),for all x∈I, lim x→{a,b}Φ(x)W(x)=lim x→{a,b}Ψ(x)W(x)=0.(1.38) (ii) The sequence (1 nP′ n)nis a sequence of monic orthogonal polynomials. Moreover the sequence (P′ n)nis orthogonal with respect to Φ(x)W(x). 1.3. Matrix Operators. 15 The matrix-valued hypergeometric equation For some particular cases of second order matrix differential operators, the polynomial eigenfunctions can be described in terms of matrix-valued hypergeometric functions. We present here two matrix-valued hypergeometric functions introduced by Tirao in [98]. These matrix-valued functions characterize, under certain restrictions on the coefficients, the vector-valued polynomial solutions around x=0 of the matrix equation x(1−x)P′′(x)+P′(x)F1(x)+P(x)F0=ΛP(x). To do so we first make clear some notation that will be used. Assume that A, B, C ∈CN×Nwith σ(C)∩{0,−1,−2,⋯}=∅. For m∈Nwe write (A, B, C)mfor the matrix function (A, B, C)0=I, (A, B, C)m+1=(A, B, C)m(B+mI)(A+mI)(C+mI)−1, and [A, B, C]mfor the matrix function [A, B, C]0=I, [A, B, C]m+1=[A, B, C]m(m2I+m(A−I)+B)(C+mI)−1. Definition 1.3.10. Let A, B, C ∈CN×Nsuch that σ(C)∩{0,−1,−2,⋯}=∅, we define (1.39) 2F1[A, B C;z]=∞ ∑ n=0 zn n!(A, B, C)n. Definition 1.3.11. Let C, U, V ∈CN×Nsuch that σ(C)∩{0,−1,−2,⋯}=∅, we define (1.40) 2H1[U, V C;z]=∞ ∑ n=0 zn n![U, V, C]n. The following theorem characterizes the vector-valued analytic solutions of certain second order matrix differential equations. Theorem 1.3.12. [98] If σ(C)∩{0,−1,−2,⋯}=∅then, 1. The function 2H1[U,V C;x]is analytic on ∣x∣<1with values in CN×N. 2. If G0∈CN, then 2H1[U,V C;x]G0is a solution of the hypergeometric equation (1.41) x(1−x)G′′ +G′(C−xU)−GV =0 such that G(0)=G0. Conversely any solution Gat x=0is of this form. In the particular case when there exist matrices Aand Bsuch that U=I+A+Band V=AB, equation (1.41) can be solved by means of the matrix-hypergeometric function (1.39). 16 CHAPTER 1. PRELIMINARIES. Theorem 1.3.13. [98] If σ(C)∩{0,−1,−2,⋯}=∅then, 1. The function 2F1[A,B C;x]is analytic on ∣x∣<1with values in CN×N. 2. If G0∈CN, then 2F1[A,B C;x]G0is a solution of the hypergeometric equation x(1−x)G′′ +G′(C−x(I+A+B))−GAB =0 such that G(0)=G0. Conversely any solution Gat x=0is of this form. Observation 1.3.14. In general and because of the different order in which the coefficients and the eigenvalues are multiplied, the matrix equation x(1−x)P′′ n(x)+P′ n(x)F1(x)+Pn(x)F0=ΛnPn(x) does not fit in the form of the previous ones. However, for a differential operator Dsuch that there exists a family of orthogonal polynomials (Pn)nwith DPn=ΛnPnand Λnis a diagonal matrix for all n≥0, then we can rewrite the previous equation as a set of N differential equations for the rows of the matrix orthogonal polynomials Pn,i, and these equations are of the form (1.41), x(1−x)P′′ n,i(x)+P′ n,i(x)(F1(x)−λiI)+Pn,i(x)F0=0. Notice that, because of Lemma 1.3.3, for a symmetric differential operator Dwith respect to a weight matrix Wwe can always find a sequence of orthogonal polynomials (Pn)nsuch that D(Pn)=ΛnPn, and Λnare diagonal matrices. 1.3.2 Matrix difference operators A finite difference operator can be expanded in terms of several basis. We can use powers of the difference operators ∆ and ∇defined by ∆(f(x))=f(x+1)−f(x),∇(f(x))=f(x)−f(x−1). Cross powers ∆i∇jare not needed since they are linear combinations of powers of ∆ and ∇(it is an easy consequence of the formula ∆∇=∆−∇). We can also use the shift operators, sl(F)(x)=F(x+l)with l∈Z. We can change from the basis ∆k,∇k,k≥0, to the basis sl, by using the formulas (k≥0) ∆k=k ∑ l=0(−1)k−l(k l)sl,∇k=k ∑ l=0(−1)l(k l)s−l, sk=k ∑ l=0(k l)∆l,s−k=k ∑ l=0(−1)l(k l)∇l. Since s−1○s1is the identity operator, all the shift operators are powers (positive or negative) of s1. We will mostly use the shift operators sl,l∈Z(see (1.33)) to define matrix difference operators, and eventually we will change to the ∆ and ∇operators. Taking into account Lemma 1.3.3, if a weight matrix Whas a symmetric second order difference operator Dsuch that for any polynomial P,D(P)is a polynomial with degree 1.3. Matrix Operators. 17 at most the degree of P, then their orthonormal polynomials (Pn)n(and then, any other sequence of orthogonal polynomials with respect to W) satisfy a second order difference equations of the form (1.42) Pn(x+1)F1(x)+Pn(x)F0(x)+Pn(x−1)F−1(x)=ΛnPn(x). The following lemma (Lemma 3.2 of [35]) characterizes the coefficients of finite difference operators, Dsuch that for every polynomial P,D(P)is a polynomial with degree at most the degree of P. Lemma 1.3.15. [35] Let Dbe a finite difference operator (1.43) D(⋅)=r ∑ l=s sl(⋅)Fl. The following conditions are equivalent (i) For any matrix polynomial P,D(P)is also a polynomial with degree at most the degree of P. (ii) The functions Fl,l=s, ⋯, r, are polynomials of degree at most r−sand dgr(r ∑ l=s lkFl)≤kfor k=0,⋯, r −s. Proceeding as we did for matrix differential operators with orthogonal polynomials as eigenfunctions, one can relate matrix orthogonal polynomials being eigenfunctions of difference operators with a discrete-discrete non-commutative bispectral problem. In the discrete case one also finds families of orthogonal polynomials being eigenfunctions of various independent difference operator. Given a weight matrix Wand a sequence of orthogonal polynomials with respect to it (Pn)n, we define the algebra D(W)by (1.44) D(W)={D=r ∑ k=s skFk(x)∣D(Pn)=ΛnPn, s <r, n ≥0}. Although we use the same notation for the algebra of difference operators and differential operators, the context will make clear which we are dealing with. Because of the left linearity of such a difference operator, it is clear that the definition does not depend on the sequence of matrix orthogonal polynomials but only on the weight. We say that a difference operator Dof the form (1.43) with Fs, Fr≠0 has order r−sand genre (r, s). Fixed a sequence of orthogonal polynomials with respect to a weight matrix W, each difference operator in Dis determined by the sequence of eigenvalues (Λn(D))n. Theorem 1.3.16. [35] Let Dbe a difference operator in D(W)with D(Pn)=Λn(D)Pn. Then, Dis determined by the sequence (Λn(D))n. Set Λ(D, n)=Λn(D), for each n≥0, the application D→Λ(D, n)is a representation of D(W)in CN×Nand the sequence of representation (Λn)nseparates elements of D(W). If D∈D(W), then the n-th eigenvalue associated to the sequence of monic orthogonal polynomials (Pn)nis a matrix polynomial in nwith degree at most the order of D. 18 CHAPTER 1. PRELIMINARIES. Theorem 1.3.17. [35] Let D=r ∑ l=s slFla difference operator such that D(Pn)=ΛnPn, where (Pn)nis the sequence of monic orthogonal polynomials with respect to W. If Fl= r−s ∑ i=0 Fi lxi, then Λn=r−s ∑ k=0(n k)r ∑ l=s lkFk l. Hence the matrix Λnis a polynomial in nof degree at most the order r−sof D. The algebra of difference operators associated to a weight matrix Wcan be endowed with a ∗-operation. Theorem 1.3.18. [35] For any D∈D(W)there is a unique difference operator D∗∈ D(W)such that ⟨D(P), Q⟩=⟨P, D∗(Q)⟩. We shall refer to D∗as the adjoint of D. The map D↦D∗is then a ∗-operation in the algebra D(W). If the operator Dhas genre (s, r)then, D∗has genre (−r, −s). By Lemma 1.3.3 it is clear that symmetric operators Dnot raising the degree of polynomials are always in D(W)and can be characterized in terms of its eigenvalues with respect to a sequence of orthonormal polynomials. If we denote by S(W)⊂D(W)the subset of symmetric difference operator, then S(W)is a real form of the space D(W), [35], D(W)=S(W)⊕iS(W). From theorem 1.3.18 it follows that a finite order difference operator D(1.43) symmetric with respect to a weight matrix and the degree of D(P)is at most the degree of Pfor all matrix polynomials Phas genre (−r, r)and order 2r. The symmetry of a finite order difference operator satisfying this condition with respect to a discrete weight matrix Wcan be guaranteed by a finite set of commuting and difference equations together with certain boundary conditions. In fact, the following theorem holds: Theorem 1.3.19. [35] For r≥0, let Dbe the finite difference operator (1.45) D(⋅)=r ∑ l=−r sl(⋅)Fl(x), where Fl(x),l=−r, ⋯, r are matrix polynomials. Let Wbe the discrete weight matrix with support S, given by (1.46) W=∑ x∈S W(x)δx. Suppose that the coefficients Fland the weight matrix Wsatisfy the following equations (1.47) Fl(x−l)W(x−l)=W(x)F∗ −l(x),for x∈(l+S)∩S,and l=0,1,⋯, r, and the boundary conditions Fl(x−l)W(x−l)=0,for x∈(l+S)/S,and l=1,⋯, r,(1.48) W(x)F∗ −l(x)=0,for x∈S/(l+S),and l=1,⋯, r.(1.49) Then, the difference operator (1.45) is symmetric with respect to W. 1.3. Matrix Operators. 19 1.3.3 Matrix q-difference operators In the whole thesis, qdenotes a real number with 0 <q<1. A q-difference operator can be expressed in terms of q-differential basic operators (q-derivatives) Dq(f(x))=f(x)−f(qx) (1−q)x, Dq−1(f(x))=f(x)−f(q−1x) (1−q−1)x or in terms of q-shift operators El(f(x))=f(qlx). For a matter of simplicity in the result exhibited here and in the foregoing chapters, we chose to study matrix q-difference operators expressed in terms of q-shift operators. In particular, and because of Lemma 1.3.3 we focus our attention in those q-difference operators taking matrix polynomials into matrix polynomials and not raising the degree, i.e D∶CN×N k[x]→CN×N k[x]. Then, we have to consider q-difference operators of the form D(⋅)=r ∑ l=s El(⋅)Fl(x), Fl(x)∈CN×N[x−1], where Fl(x)satisfy some degree conditions, as it is shown in the following theorem, that is a straightforward adaptation of Theorem 3.2 [35] Theorem 1.3.20. Let D=r ∑ l=s El(⋅)Fl(x),(1.50) with r, s integers such that s≤r. The following conditions are equivalent: (i) D∶CN×N k[x]→CN×N k[x]for all k≥0. (ii) Fl(x)∈CN×N r−s[x−1]for l=s,...,r and ∑r l=sqlkFl(x)∈CN×N k[x−1]for k=0,...,r−s. We say that a q-difference operator of the form (1.50) has order r−sand genre r, s). It happens that if a q-difference operator is symmetric with respect to a weight matrix, then s=−r. The following theorem establishes the symmetry equations and boundary condition that assure the symmetry of a q-difference operator with order 2rand genre (r, −r), with respect to a q-weight. Theorem 1.3.21. For r≥0, let Dbe the q-difference operator (1.51) D=r ∑ l=−r El(⋅)Fl(x) where Fl(x)are matrix polynomials in x−1. Let Wbe a q-weight given by W=∞ ∑ x=0 W(qx)δqx. 20 CHAPTER 1. PRELIMINARIES. Suppose that the coefficients Fland the q-weight matrix Wsatisfy the following equations F0(qx)W(qx)=W(qx)F0(qx)∗, x ∈N,(1.52) Fl(qx−l)W(qx−l)=qlW(qx)F−l(qx)∗, x ≥l(1.53) and the boundary conditions F−l(qx)W(qx)=0,forx=0,⋯, l −1,andl=1,⋯, r(1.54) lim x→∞q2xFl(qx)W(qx)→0,forl=1,⋯, r lim x→∞qx(Fl(qx)W(qx)−W(qx)Fl(qx)∗)→0,forl=1,⋯, r. Then, the q-difference operator Dis symmetric with respect to W. Proof. Let Dbe a q-difference operator of the form (1.51) and Waq-weight such that the symmetry equations (1.52) and (1.53) as well as the boundary conditions (1.54) hold. For Mlarge enough (much bigger that r) we consider the truncated inner product ⟨P, Q⟩M=M ∑ x=0 qxP(qx)W(qx)Q(qx)∗. It is clear that for any two matrix polynomials P, Q ∈CN×N[x] lim M→∞⟨P, Q⟩M=⟨P, Q⟩. We are going to prove that ⟨D(P), Q⟩M=⟨P, D(Q)⟩M+Θ(M, x) where Θ(M, x)is a matrix function such that limM→∞Θ(M, x)=0. Let P, Q ∈CN×N[x]. Then, ⟨D(P), Q⟩M=M ∑ x=0 r ∑ k=−r P(qk+x)Fk(qx)W(qx)Q(qx)∗qx (1.55) =M ∑ x=0 r ∑ k=1 P(qx−k)F−k(qx)W(qx)Q(qx)∗qx+M ∑ x=0 P(qx)F0(qx)W(qx)Q(qx)∗qx +M ∑ x=0 r ∑ k=1 P(qk+x)Fk(qx)W(qx)Q(qx)∗qx. Taking into account the symmetry equation (1.53) we can write the first factor on the previous sum as r ∑ k=1 M ∑ x=0 P(qx−k)F−k(qx)W(qx)Q(qx)∗qx=r ∑ k=1 M ∑ x=0 P(qx−k)qkF−k(qx)W(qx)Q(qx)∗qx−k =r ∑ k=1 M ∑ x=k P(qx−k)W(qx−k)Fk(qx−k)∗Q(qx)∗qx−k +r ∑ k=1 k−1 ∑ x=0 P(qx−k)F−k(qx)W(qx)Q(qx)∗qx. 2.2. Examples. 27 Proof. For this example we have that κis infinite and F−1=(I+A)−1x,(2.11) F0=−J−(I+A)−1x,(2.12) F1=a(I+A).(2.13) We now use Theorem 1.3.19 for r=1 to prove that the operator Dis symmetric. We have to check the boundary condition W(0)F∗ −1(0)=0 and the equations F1(x−1)W(x−1)=W(x)F∗ −1(x),for x=1,2,⋯ F0(x)W(x)=W(x)F∗ 0(x),for x=0,1,⋯. We proceed in three steps. First Step. Boundary condition W(0)F∗ −1(0)=0. Since F−1(x)=(I+A)−1x, we have F−1(0)=0 and the boundary condition follows straightforwardly. Second Step. F1(x−1)W(x−1)=W(x)F∗ −1(x)for x=1,2,⋯. We use Theorem 2.1.1 to prove that F1,F−1and Wsatisfy this first order difference equation. We first check that for s(x)=√ax the matrix polynomials F−1and F1satisfy F1(x−1)F−1(x)=s2(x)I, x ≥1. But this is straightforward from the definition of F−1and F1(see (2.11) and (2.13)). We then factorize the matrix weight Win the form W=∑∞ x=0T(x)T∗(x)δx,where T(x)=√ax x!(I+A)x. Since T(0)=I, we only have to check that for x≥1, T(x−1)=F−1(x) s(x)T(x). Indeed, we have F−1(x) s(x)T(x)=(I+A)−1x √ax √ax x!(I+A)x=√x a√ax x!(I+A)−1(I+A)x =¿ Á Á Àax−1 (x−1)!(I+A)x−1=T(x−1). Theorem 2.1.1 gives now that F1,F−1and Wsatisfy F1(x−1)W(x−1)=W(x)F∗ −1(x) for x=1,2,⋯. Third Step. F0(x)W(x)=W(x)F∗ 0(x)for x∈N. Since W(x)=T(x)T∗(x)the equation F0W=WF ∗ 0is straightforwardly equivalent to the Hermiticity of the function χ(x)=T−1(x)F0(x)T(x), x ∈N. 28 CHAPTER 2. DISCRETE ORTHOGONAL POLYNOMIALS We now explicitly compute the function χto see that actually it is a real diagonal matrix function. To do that, we expand the function χ(x)in a power series. By writing S= log (I+A)we get (I+A)x=eSx. Taking into account that Aand Scommute, and using formula (1.24), we have χ(x)=T−1(x)F0T(x)=e−Sx (−J−(I+A)−1x)eSx =∞ ∑ i=0(−1)i+1 i!adi SJxi−(I+A)−1x. To compute adi SJ,i≥1, we use Lemma 2.2.2 and we get adSJ=[S, J]=(I+A)−1−I. For adi SJ,i≥2, since Scommutes with A, we have adi SJ=adi−1 S(adSJ)=adi−1 S((I+A)−1−I)=0. Hence, we get χ(x)=−J−xI. That is, the matrix function χis real diagonal, and then Hermitian. Thus, Theorem 1.3.19 for r=1 shows that the second order operator D(3.5) is symmetric with respect to W. We now use Lemma 1.3.3 to prove that the orthogonal polynomials are common eigenfunctions of the operator D. Since we have already proved that Dis symmetric with respect to W, it is enough to check that for any polynomial Pthe degree of D(P)is at most the degree of P. But that is a consequence of Lemma 1.3.15 taking into account that dgr(Fi)≤2,for i=0,1,−1, dgr(F1−F−1)=dgr(a(I+A)−(I+A)−1x)=1, dgr(F0+F1+F−1)=dgr(−J+a(I+A))=0. The expression for the eigenvalues follows by Theorem 1.3.17. This completes the proof. 2.2.2 Example with dgr(F1)=1and unbounded support. Our second example can be considered as a matrix relative of the Meixner scalar weight. It has unbounded support and the coefficient F1of its associated second order difference operator is a polynomials of degree 1. Theorem 2.2.3. Let Aand Jbe the N×Nnilpotent and diagonal matrices, respectively, given by (1.25). Given two positive real numbers aand c, with 0<a<1, we consider the matrix (2.14) RA,a =(I−A)(I−aA)−1, and the weight matrix defined by (2.15) W=1 Γ(c)∞ ∑ x=0 axΓ(x+c) x!Rx A,a (R∗ A,a)xδx. 2.2. Examples. 29 Then, the second order difference operator (2.16) D(⋅)=s1(⋅)F1(x)+s0(⋅)F0(x)+s−1(⋅)F−1(x), where F−1(x)=R−1 A,ax,(2.17) F0(x)=(a−1)J−x(aRA,a +R−1 A,a),(2.18) F1(x)=a(x+c)RA,a,(2.19) is symmetric with respect to the weight matrix W. Moreover, the monic orthogonal polynomials with respect to Ware common eigenfunctions of D, with eigenvalues given by Λn=(a−1)J+acRA,a +n(aRA,a −R−1 A,a). Proof. We use again Theorem 1.3.19 for r=1 to prove that the operator Dis symmetric. We have to check the boundary condition W(0)F∗ −1(0)=0 and the equations F1(x−1)W(x−1)=W(x)F−1(x),for x=1,2,⋯,(2.20) F0(x)W(x)=W(x)F∗ 0(x),for x=0,1,⋯.(2.21) We omit the proofs of the boundary condition and the equation (2.20) because they are similar to those in Steps 1 and 2 in Theorem 2.2.1 (just taking here s(x)=√a(x+c−1)x). To prove F0(x)W(x)=W(x)F∗ 0(x)for x∈N, we proceed as follows. We first write W(x)=T(x)T∗(x), where T(x)=¿ Á Á ÀaxΓ(x+c) Γ(c)x!Rx A,a. As we have already pointed out, the equation F0(x)W(x)=W(x)F∗ 0(x)is then equivalent to the Hermiticity of the matrix function χ(x)=T−1(x)F0(x)T(x). We are going to show that this is, in fact, a real diagonal matrix. Consider now the analytic function at 0 f(z)=log (1−z 1−az ), and write S=f(A). This gives eS=RA,a and e−S=R−1 A,a, and hence χ(x)=T−1(x)F0(x)T(x) =e−Sx ((a−1)J−aeSx−e−Sx)eSx =∞ ∑ i=0(−1)i i!(a−1)adi SJxi−aeSx−e−Sx. We now compute explicitly the function χ Using Lemma 2.2.2, we see that −(a−1)[S, J]−aeS−e−Sis, in fact, a multiple of the identity matrix. Indeed, since f′(z)=a 1−az −1 1−z, 30 CHAPTER 2. DISCRETE ORTHOGONAL POLYNOMIALS according to this lemma, we have [S, J]=A(1−A)−1−aA(1−aA)−1. On the other hand, a simple computation gives −aeS−e−S=−(a+1)I−a(a−1)A(1−aA)−1+(a−1)A(I−A)−1. Then, we get −(a−1)[S, J]−aeS−e−S=−(1+a)I. For the rest of the coefficients, we proceed as in Theorem 2.2.1 to get adi RJ=0 for i≥2. We have then that χ(x)=(a−1)J−(1+a)Ix. This is a real diagonal matrix for all x≥0, and so Hermitian. Then, F0(x)W(x)= W(x)F∗ 0(x)holds for x∈N. Using Theorem 1.3.19 for r=1 we can conclude that the second order difference operator (2.16) is symmetric with respect to the weight matrix W(2.15). We now use Lemma 1.3.3 to see that the matrix orthogonal polynomials with respect to Ware common eigenfunctions of the difference operator (2.16). Since we have already proved the symmetry of D, we have just to see that for any polynomial P,D(P)has degree at most the degree of P. This is a consequence of Lemma 1.3.15 since dgr(Fi)≤2,for i=0,1,−1, dgr(F1−F−1)=dgr(a(x+c)RA,a −R−1 A,ax)=1, dgr(F0+F1+F−1)=dgr((a−1)J+acRA,a)=0. The expression for the eigenvalues follows by Theorem 1.3.17. The proof of the theorem is now complete. 2.2.3 Example with dgr (F1)=1and finite support Our third example can be considered as a matrix relative of the Krawtchouk scalar weight. It has then finite support and the coefficient F1of its associated second order difference operator is a polynomial of degree 1 (the proof will be omitted because is similar to that of theorems 2.2.1 and 2.2.3). Theorem 2.2.4. Let Aand Jbe the N×Nmatrices given by (1.25). Given a positive integer κand a positive real number a, we consider the matrix (2.22) RA,a =(I+A)(I−aA)−1, and the matrix weight defined by (2.23) W=κ ∑ x=0 axΓ(κ+1) Γ(κ+1−x)x!Rx A,a (R∗ A,a)xδx. 2.2. Examples. 31 Then, the second order difference operator (2.24) D(⋅)=s1(⋅)F1(x)+s0(⋅)F0(x)+s−1(⋅)F−1(x), where F−1(x)=R−1 A,ax, F0(x)=−(a+1)J+aRA,ax−R−1 A,ax, F1(x)=−aRA,ax+aκRA,a, is symmetric with respect to W. Moreover, the monic orthogonal polynomials with respect to Ware common eigenfunctions of Dand its eigenvalues are given by Λn=−(a+1)J+aκRA,a −n(aRA,a +R−1 A,a). 2.2.4 Example with dgr(F1)=2 Our third example seems not to have any scalar relative. The weight matrix has bounded support and the difference coefficient F1has degree two. For a positive integer κand complex numbers vi∈C, 1 ≤i≤N−1, we define the nilpotent matrices (with order of nilpotency 2 and N−1 respectively) B=−2v1 κ+1E1,2,(2.25) C=v2E1,3+N−1 ∑ i=1 i≠2 viEi,i+1.(2.26) We also define the diagonal matrix (2.27) L=E1,1+5E2,2+N ∑ i=3(2i−3)Ei,i. It is a matter of computation to check that these matrices satisfy the following equations B2=0,(2.28) BC =CB =0,(2.29) adCL=2C−(κ+1)B,(2.30) adCkL=2kCk,for k≥2,(2.31) adBL=4B.(2.32) We are now ready to introduce our last example. Theorem 2.2.5. Let B,Cand Lbe the matrices defined by (2.25), (2.26) and (2.27) respectively. We consider the matrix (2.33) RC=(I−1 2C)(I+1 2C)−1 32 CHAPTER 2. DISCRETE ORTHOGONAL POLYNOMIALS and the weight matrix defined by (2.34) W=κ ∑ x=0 Γ(κ+1) Γ(κ+1−x)x!(Rx C−Bx(x+1) 2)((R∗ C)x−B∗x(x+1) 2)δx. Then, the second order difference operator (2.35) D(⋅)=s1(⋅)F1(x)+s0(⋅)F0(x)+s−1(⋅)F−1(x), where F1=Bx2+((1−κ)B−RC)x+κ(RC−B),(2.36) F0=−2Bx2−(R−1 C+(1−κ)B−RC)x+L,(2.37) F−1=Bx2+R−1 Cx,(2.38) is symmetric with respect to the weight matrix W (2.34). Moreover, the monic orthogonal polynomials with respect to Ware common eigenfunctions of Dand its eigenvalues are given by Λn=L+κ(RC−B)+n((1−κ)B−RC−R−1 C)+n(n−1)B. Proof. We use Theorem 1.3.19 for r=1 to prove the symmetry of Dwith respect to W. For this we have to check the boundary conditions, F1(κ)W(κ)=0 and W(0)F∗ −1(0)=0 and the equations F1(x−1)W(x−1)=W(x)F∗ −1(x),for x=1,⋯, κ, F0(x)W(x)=W(x)F∗ 0(x),for x=0,1,⋯, κ. We skip over the proof of the boundary conditions and the first symmetry equation since it is similar to that for Theorem 2.2.1 (taking now s(x)=√x(κ+1−x)). We write W=∑κ x=0T(x)T∗(x)δxfor T(x)=¿ Á Á ÀΓ(κ+1) Γ(κ+1−x)x!(Rx C−Bx(x+1) 2). Then, the equation F0(x)W(x)=W(x)F∗ 0(x)is equivalent to the Hermiticity of the matrix function χ(x)=T−1(x)F0(x)T(x). We again compute explicitly the function χto see that it is in fact a real diagonal matrix. Consider the analytic function at 0 (2.39) f(z)=log (1+z 2 1−z 2), and write S=f(C). This gives R−1 C=eSand RC=e−S. The function f(C)only contains odd powers of C, then by the property (2.29) we get that SB =BS =0, and so eSxB=BeSx =B=e−SxB=Be−Sx. 2.2. Examples. 33 Taking into account this remark as well as the property (2.28) we can write T−1(x)=¿ Á Á ÀΓ(κ+1−x)x! Γ(κ+1)(e−Sx −Bx(x+1) 2)−1 =¿ Á Á ÀΓ(κ+1−x)x! Γ(κ+1)((I−Bx(x+1) 2)e−Sx)−1 =¿ Á Á ÀΓ(κ+1−x)x! Γ(κ+1)(I+Bx(x+1) 2)eSx. Because of (2.28) and (2.32), we have adi BL=0 for i≥2. Taking into account all these properties we can write χ(x)as the power series χ(x)=L+((κ−1)B+e−S−eS+adSL+1 2adBL)x(2.40) +(−2B+1 2adB(L+adSL)+1 2ad2 SL)x2 +∞ ∑ i=3(1 2adB(adi−1 SL (i−1)!+adi−2 SL (i−2)!)+adi SL i!)xi. We now prove that, except the first one, all the coefficients above vanish. To do that, we use the following lemma. Lemma 2.2.6. Let B,Cand Lbe the matrices defined by (2.25), (2.26) and (2.27). For an analytic function fat 0 the following equations hold ((κ+1)f′(0)−2)B+adf(C)L+1 2adBL=2Cf′(C),(2.41) adi+1 f(C)L=0,for i≥1,(2.42) adB(adf(C)L)=0.(2.43) Proof. Write f(z)=∑k≥0akzk. Taking into account equations (2.30) and (2.31), we get adf(C)L=∑ k≥0 ak[Ck, L]=2∑ k≥0 akkCk−(κ+1)a1B=2Cf′(C)−(κ+1)f′(0)B, and then, using equation (2.32) we get ((κ+1)f′(0)−2)B+adf(C)L+1 2adBL=((κ+1)f′(0)−2)B+2Cf′(C) −(κ+1)f′(0)B+2B=2Cf′(C). Since BC =CB =0, any power of Ccommute with Bthen, we have adi+1 f(C)L=adi f(C)(2Cf′(C)−(κ+1)a1B)=0,for i≥1, adB(adf(C)L)=adB(2Cf′(C)−(κ+1)a1B)=0. 34 CHAPTER 2. DISCRETE ORTHOGONAL POLYNOMIALS Using now equation (2.41) of Lemma 2.2.6 for the function fin (2.39) (for which f′(0)=1) and S=f(C), we get (κ−1)B+e−S−eS+adSL+1 2adBL=e−S−eS+2Cf′(C)(2.44) =e−S−eS+((I+C 2)−1+(I−C 2)−1 ) =0. On the other hand, using equations (2.42) and (2.43) of Lemma 2.2.6, we obtain −2B+1 2adB(L+adSL)+1 2ad2 SL=0,(2.45) 1 2adB(adi−1 SL (i−1)!+adi−2 SL (i−2)!)+adi SL i!=0, i ≥3,(2.46) where we have also used the equation (2.32). Equations (2.40), (2.44), (2.45) and (2.46) then give that χ(x)=L, which it is a real diagonal matrix. So F0(x)W(x)=W(x)F0(x)for x=0,1,⋯, κ. By Theorem 1.3.19 for r=1, we conclude that Dis symmetric with respect to W. To see that the orthogonal polynomials with respect to Ware common eigenfunctions of D, we use Theorem 1.3.3. Since we have already proved that Dis symmetric with respect to W, it remains to see that for any polynomial P,D(P)is a polynomial with degree at most the degree of P. This is a consequence of Lemma 1.3.15, since dgr(Fi)≤2 for i=−1,0,1, dgr(F1−F−1)=dgr(((1−κ)B−e−S−eS)x+κ(e−S−B))=1, dgr(F0+F1+F−1)=dgr(L+κ(e−S−B))=0. Finally to obtain the expression for the eigenvalues we apply Theorem 1.3.17 3.2. Examples. 43 In our example we have that (n 2)F1,2+nF1,1−nF−1,1+G0=(n+1−a−nb −ab 0n−a), G∗ 0=(1−a−ab 0−a). For a fixed n≥2, the matrices (n 2)F1,2+nF1,1−nF−1,1+G0and G∗ 0do not share any eigenvalue. This implies that the equation (3.11) defines µn,n≥2, in an unique way from µ0and µ1(see [61], p. 225). But µ1has to be Hermitian, since Uis a weight matrix, hence it is easy to see that equation (3.11) for n=1 also defines µ1in an unique way from µ0. It is just a matter of calculation to see that the set of solutions µ0of the equation (3.10) is formed by the matrices (η abγ abγ γ ), where γ>0 and η>γa2b2(since we are assuming that µ0is the first moment of a weight matrix, and so it has to be a positive definite matrix). On the other hand, the first moment of γWξ,γ>0, ξ ∈R, is γea(ξ+ab2+a2b2ab ab 1). By choosing ξ=η/γ−ab2−a2b2, we see that for each weight matrix U∈Υ(D)there exist γ>0 and ξ∈Rsuch that Uhas the same first moment µ0as γe−aWξ. Hence, since the (generalized) moments µn,n≥1, of Uand γWξare uniquely defined from µ0, we conclude that Uand γWξhave the same generalized moments µn,n≥0. It is then clear that U and γWξhave also the same moments sn,n≥0, where sn=∫xndU(x). We now prove that actually U=γWξand then, ξ>0 (since Uis a weight matrix). This can be done by different approaches. One of them uses Fourier transform and it works as follows: consider the Fourier transform F(X)of a weight matrix Xdefined by F(X)(x)=∫ReitxdX(t),x∈R.F(X)is a matrix C∞-function in Rand (d dx)n(F(X)(x))=∫R(it)neitxdX(t), x ∈R. This shows that the moments of Xare just (d/dx)n(F(X))(0)/in,n≥0. Since Uand γWξhave the same moments sn,n≥0, we have that the Fourier transforms F(U)of Uand F(γWξ)of γWξhave at 0 the same derivatives of any order. These derivatives are equal to insn,n≥0. It is easy to see that the Fourier transform F(γWξ)is actually a matrix entire function, and accordingly lim n→∞∣(sn)i,j∣rn n!=0, for all r>0, where Yi,j denotes the (i, j)entry of a matrix Y. Since sn,n≥0, are the moments of the weight matrix Uand the even moments s2nof a weight matrix are positive definite matrices, we conclude that lim n→∞Tr(s2n)r2n (2m)!=0. 44 CHAPTER 3. THE CONVEX CONE. Since Uand γWξhave the same moments, according to Lemma 2.1 of [36], there exists an entire function Φ such that Φ(x)=F(U)(x),x∈R. Hence, the corresponding Fourier transforms of Uand γWξare entire functions which have at 0 the same derivatives of any order. That is, the entire functions Φ and F(γWξ)are equal and then, F(γWξ)(x)= F(U)(x),x∈R. So U=γWξ. As we wrote in the previous Section, the assumptions in the Lemma 3.1.1 are not so restrictive as at a first glance appeared to be. In fact, all the examples of weight matrices constructed in Chapter 2 satisfy that assumptions. For the sake of completeness, we display a second example (the proof is omitted), now in arbitrary size, using the same method and based again in an example from Chapter 2. In this case the convex cone Υ(D)is, at least, N-dimensional. For this example we recover the matrices, A,Jin (1.25) and introduce the N×N matrix RA,a: (3.12) A=N−1 ∑ i=1 viEi,i+1, J =N ∑ i=1(N−i)Ei,i, RA,a =(I−A)(I−aA)−1, where symbol Ei,j stands for the N×Nmatrix with entry (i, j)equal to 1 and 0 otherwise, and v1,⋯, vN−1are complex numbers. Theorem 3.2.3. For real numbers aand cwith 0<a<1and c>0let Dbe the second order difference operator defined by D(⋅)=s1(⋅)F1(x)+s0(⋅)F0(x)+s−1(⋅)F−1(x), where F−1(x)=R−1 A,ax, F1(x)=a(x+c)RA,a, F0(x)=(a−1)(J−(N−1)I)−x(aRA,a +R−1 A,a). Then, {γWξ1,⋯,ξN−1∶γ, ξi>0, i =1,⋯, N −1}⊆Υ(D)where Wξ1,⋯,ξN−1=1 Γ(c)∞ ∑ x=0 axΓ(x+c) x!Rx A,a ⎛⎜⎜⎜⎝ ξ10⋯0 ⋮ ⋱ ⋱ ⋮ 0⋯ξN−10 0⋯0 1 ⎞⎟⎟⎟⎠(R∗ A,a)xδx. 3.2.2 Examples constructed by using the second method We now use the second method to study the convex cone of some second order difference operators. We first consider the second order difference operator (3.13) D(⋅)=s1(⋅)F1(x)+s0(⋅)F0(x)+s−1(⋅)F−1(x). 3.2. Examples. 45 where F−1, F0, F1are the matrix polynomials given by F−1(x)=(−b(x−1)(a−1)ab2x(x−1)(a−1) −1xba ), F0(x)=(b(a(c+x)−x)−b−1−ax (b2(c+2x)(a−1)−1) 1−2xba ), F1(x)=(0a(c+x)((x+1)b2(a−1)−1) 0b(c+x)a), along with the weight matrix W=∞ ∑ x=0 axΓ(x+c) x!(x+b2(a−1)2x2b(a−1)x b(a−1)x1)δx. Theorem 3.2.4. For real numbers a,band cwith 0<a<1,0<c,a/=1/2+1/(2b2)and b/=0, let Dbe the second order difference operator defined by (3.13). Then, Υ(D)={γW +ξ(a∣b∣c−sign(b) −sign(b)1/(a∣b∣c))δ−c∶γ>0, ξ ≥0}, where sign(b)stands for the sign of b. Proof. It is a matter of calculation to check that Wand Dsatisfy the symmetry equations F1(x−l)W(x−l)=W(x)F∗ −1(x),for x≥1 F0(x)W(x)=W(x)F∗ 0(x),for x≥0, as well as the boundary condition W(0)F∗ −1(0)=0. By Theorem 1.3.19 we can conclude that W∈Υ(D). To simplify the writing, we set M=(a∣b∣c−sign(b) −sign(b)1/(a∣b∣c)). For x=−cwe then, have F−1(−c)M=(b(c+1)(a−1)acb2(c+1)(a−1) 1−cba )( a∣b∣c−sign(b) −sign(b)1/(a∣b∣c))=0, F1(−c)M=(0 0 0 0)( a∣b∣c−sign(b) −sign(b)1/(a∣b∣c))=0, F0(−c)M=(−1+cb2 b−ac (cb2a−cb2+1) 1 2cba )( a∣b∣c−sign(b) −sign(b)1/(a∣b∣c)) =(sign(b)c2a2b2−a∣b∣c −a∣b∣c−sign(b)), MF∗ 0(−c)=(a∣b∣c−sign(b) −sign(b)1/(a∣b∣c))( −1+cb2 b1 −ac (cb2a−cb2+1)2cba) =(sign(b)c2a2b2−a∣b∣c −a∣b∣c−sign(b)). 46 CHAPTER 3. THE CONVEX CONE. By Theorem 3.1.2, the weight matrix γW +ξMδ−c,γ>0, ξ ≥0, is also in the convex cone Υ(D)of D. We now prove that actually this family of weight matrices provides all the weight matrices in the convex cone Υ(D), that is Υ(D)={γW +ξMδ−c∶γ>0, ξ ≥0}. The proof is similar to that of Theorem 3.2.2, hence we only sketch it. Lemma 3.1.3 gives that the generalized moments µn=∫(x n)dU(x),n≥0, of each weight matrix U∈Υ(D)have to satisfy the moment equations (3.3) and (3.4). If we write F−1(x)=F−1,2(x 2)+F−1,1(x 1)+F−1,0, F1(x)=F1,2(x 2)+F1,1(x 1)+F1,0, G0=F−1+F0+F1, these moment equations are, for n=0, (3.14) G0µ0=µ0G∗ 0, and for n≥1 [(n 2)F1,2+(n 1)F1,1−(n 1)F−1,1+G0]µn (3.15) +[(n−1 2)F1,2+(n−1 1)F1,1+F1,0]µn−1−F−1,0 n−1 ∑ j=0(−1)j+n−1µj=µnG∗ 0. In our example, (n 2)F1,2+nF1,1−nF−1,1+G0and G0are upper triangular matrices with diagonal entries equal to b(a−1)n+abc +ab −b−1/b, abc −ab, abc +ab −b−1/b, abc, respectively. Hence, for a fixed n≥1, and except for the values a=1/2+1/(2b2)and a=1+1/((n+1)b2), the matrices (n 2)F1,2+nF1,1−nF−1,1+G0and G∗ 0do not share any eigenvalue. We need not to consider these exceptional values: a=1+1/((n+1)b2), since we assume a<1, and a≠1/2+1/(2b2). Hence, equation (3.15) defines µn,n≥1, in an unique way from µ0(see [61], p. 225). It is just a matter of calculation to see that the set of solutions µ0of the equation (3.14) is formed by the matrices (η−abcτ −abcτ τ ), where τ>0 and η>τa2b2c2(since we are assuming that µ0is the first moment of a weight matrix, and so it has to be a positive definite matrix). On the other hand, the first moment of γW +ξMδ−c,γ, ξ ∈R, is γΓ(c) (1−a)c⎛⎜⎝ ac(a2b2c−ab2(c−1)−1−b2) a−1−abc −abc 1⎞⎟⎠+ξM. It is then easy to see that given τand ηthere exist γ, ξ ∈Rsuch that Uhas the same first moment µ0as γW +ξMδ−c. Hence, since the generalized moments µn,n≥1, of Uand 3.2. Examples. 47 γW +ξMδ−care uniquely defined from µ0, we conclude that Uand γWξhave the same generalized moments µn,n≥0. It is then clear that Uand γW +ξMδ−chave also the same moments sn,n≥0, where sn=∫xndU(x). In order to prove that actually U=γW +ξMδ−c, we can use again the Fourier transform (as in the proof of Theorem 3.2.2), taking into account that F(U)and F(γW +ξMδ−c) have at 0 the same derivatives of any order, that F(γW +ξMδ−c)is an analytic function in the open half plane {z∈C∶log a<Iz}and that 0 is contained in that half plane (since 0<a<1). We finally display an example in arbitrary size N×N. Consider again the N×Nmatrices given by (1.25), A=N−1 ∑ i=1 viEi,i+1, J =N ∑ i=1(N−i)Ei,i, where the parameters visatisfy the following constrains: for i=1,⋯, N −2, (3.16) (N−i−1)a∣vi∣2∣vN−1∣2+(N−1)∣vi∣2−i(N−i)∣vN−1∣2=0. Let Wbe the discrete weight matrix given by (3.17) W=∞ ∑ x=0 ax x!(I+A)x(I+A∗)xδx. It was shown in Theorem 2.2.1 that the difference operator D1(⋅)=s−1(⋅)(I+A)−1x+s0(⋅)(−J−(I+A)−1x)+as1(⋅)(I+A) is symmetric with respect to the weight matrix (3.17). In [57, Theorem 3] it was proved that, under the constrains (3.16), the difference operator D2(⋅)=s−1(⋅)H−1(x)+s0(⋅)H0(x)+s1H1(x) where H−1(x)=[(I+A)−1−I]x2+(N−1 a∣vN−1∣2I+J)x, H1(x)=[(I+A)−1−I]x2+(2J−NI −aA +(I+A)−1)x +a(I+A)( N−1 a∣vN−1∣2I+J)(I+A∗), H0(x)=−H−1(x)−H1(x), is also symmetric with respect to W. Theorem 3.2.5. Let wbe the non zero (row) vector in CNwhose entries satisfy (3.18) wi+1=(N−i) a¯vi wi, w1=1, 48 CHAPTER 3. THE CONVEX CONE. and write M0for the N×Npositive semidefinite matrix M0=w∗w. For λ=−(N−1)(1+a∣vN−1∣2) a∣vN−1∣2, consider the operator D3=λD1+D2. Then, {γW +ξM0δ0∶γ>0, ξ ≥0}⊆Υ(D3). Proof. Since D3is a linear combination of two symmetric operators with respect to W, it is also symmetric with respect to W. Write D3(⋅)=s−1(⋅)F−1(x)+s0(⋅)F0(x)+s1F1(x). By Lemma 3.1.2, if we prove that D3satisfies the conditions (3.19) F1(0)M0=0, M0F∗ −1(0)=0, F0(0)M0=M∗ 0F∗ 0(0), then, we have that D3is symmetric with respect to the weight matrices γW +ξM0δ0,γ>0 and ξ≥0, i.e. γW +ξM0δ0∈Υ(D3). The coefficients of D3are given by F1(x)=[(I+A)−1−I]x2+[2J−NI −aA +(I+A)−1]x +a(I+A)[λI +(N−1 a∣vN−1∣2I+J)(I+A∗)], F−1(x)=[(I+A)−1−I]x2+[N−1 a∣vN−1∣2I+J+λ(I+A)−1]x, F0(x)=−F−1(x)−F1(x)−λ(J−a(I+A)). It is obvious that the second condition in (3.19) holds since F−1(0)=0. To see that the first condition in (3.19) holds, we point out that under the constrains (3.16) we have F1(0)=a(I+A)[λI +(N−1 a∣vN−1∣2I+J)(I+A∗)] =a(I+A)[−(N−1)I+J+N−1 a∣vN−1∣2A∗+JA∗] =a(I+A)[N ∑ i=1(1−i)Ei,i +N ∑ i=2((N−1)¯vi−1 a∣vN−1∣2+(N−i)¯vi−1)Ei,i−1] =a(I+A)[N ∑ i=1(1−i)Ei,i +N ∑ i=2(i−1)(N−i+1) avi−1Ei,i−1], where the last equality holds because of the constrains on vi, (3.16). Now it is straightforward to see that, because of (3.18), w∗is in the kernel of F1(0), and hence F1(0)M0=0. For the third condition in (3.19), notice that F0(0)M0=(−F1(0)−F−1(0)−λ(J−a(I+A)))w∗w=−λ(J+λa (I+A))w∗w. 3.2. Examples. 49 And also M0F∗ 0(0)=−λw∗w(J−a(I+A∗)).Then, to see that F0(0)M0=M∗ 0F∗ 0(0) it suffices to check the following equality, (−J+aA)w∗w=w∗w(−J+aA∗). We write w=N ∑ i=0 wiei, where (ei)N i=1is the canonical base of CN. Then, we have that (−J+aA)M0=(N ∑ i=1(i−N)Ei,i +a N ∑ i=1 viEi,i+1)w∗w =(N ∑ i=1((i−N)¯wi+avi¯wi+1)ei)w =N ∑ i,j=1((i−N)¯wiwj+avi¯wi+1wj)Ei,j , M0(−J+aA∗)=N ∑ i,j=1((j−N)¯wiwj+a¯vj¯wiwj+1)Ei,j. So we have that (−J+aA)M0=M0(−J+aA∗)if and only if (i−N)¯wiwj+avi¯wi+1wj=(j−N)¯wiwj+a¯vj¯wiwj+1. By dividing both sides of the previous equality by ¯wiwjwe get that (−J+aA)M0= M0(−J+aA∗)if and only if (i−N)+avi ¯wi+1 ¯wi=(j−N)+a¯vj wj+1 wj . But this is true since by (3.18)¯wi+1 ¯wi=(N−i) avi , and then both sides of the previous equality are, indeed, 0. With this last equality it is proved the third condition in (3.19). Chapter 4 Matrix polynomials satisfying q-difference equations. In the theory of scalar orthogonal polynomials, those families of orthogonal polynomials being solutions of a q-differential equation of the form (1.29) σ(x)Dq−1(Dq(p(x)))+τ(x)Dq(p(x))=λp(x), where σand τare polynomials and Dqis the q-differential operator (1.30), are called q-classical polynomials of the Hahn class (we will refer to them simply as q-classical polynomials). The previous q-differential equation can be written in terms of q-shift operators, El(f(x))=f(qlx)for l∈Z, by (4.1) f−1(x)E−1p(x)+f0(x)E0p(x)+f1(x)E1p(x)=λp(x), where f−1(x)=−σ(x) (1−q2)x2f0(x)=(1+q−1)σ(x) (1−q)(1−q−1)x2+τ(x) (1−q)x, f1(x)=−σ(x) (1−q)(1−q−1)x2−τ(x) (1−q)x. We consider here a matrix analogue of the q-difference operator, defined by E−1(⋅)F−1(x)+E0(⋅)F0(x)+E1(⋅)F1(x), where Fi(x)are matrix polynomials in x−1such that dgr(Fi)≤2, dgr(qF1+F0+q−1F−1)≤1 and dgr(F1+F0+F−1)=0. Such a q-difference operator is symmetric with respect to a q-weight if it satisfies the symmetry equations F0(qx)W(qx)=W(qx)F0(qx)∗, x ∈N,(4.2) F1(qx−1)W(qx−1)=qW (qx)F−1(qx)∗, x ∈N/{0},(4.3) 51 52 CHAPTER 4. MATRIX q-POLYNOMIALS. and the boundary conditions W(1)F−1(1)∗=0,(4.4) q2xF1(qx)W(qx)→0,as x→∞,(4.5) qx(F1(qx)W(qx)−W(qx)F1(qx)∗)→0,as x→∞.(4.6) In this chapter we introduce a method to solve the symmetry equation (4.3) for a second-order q-difference operator with respect to a q-weight (4.7) W=∞ ∑ x=0 qxW(qx)δqx. We then construct a matrix relative of the little q-Jacobi polynomials and we explore some of their properties. The following lemma give us a method to solve equation (1.53) in Theorem 1.3.21. We omit the proof for being completely analogous to that of Theorem 2.1.1. Lemma 4.0.6. Let s(x)be a scalar function satisfying s(qx)≠0for x∈N/{0}. Assume that F1and F−1are matrix-valued polynomials such that F1(qx−1)F−1(qx)=q∣s(qx)∣2I, ∀x∈N/{0}.(4.8) Let Tbe a solution of the q-difference equation T(qx−1)=s(qx)−1F−1(qx)T(qx), x ∈N/{0}, T(1)=I.(4.9) Then, the q-weight defined by W(qx)=T(qx)T(qx)∗satisfies the symmetry equation F1(qx−1)W(qx−1)=qW (qx)F−1(qx)∗, x ∈N/{0}. 4.1 A matrix-valued relative of the little q-Jacobi polynomials In this section we construct a family of matrix orthogonal polynomials that can be considered as a matrix relative of the little q-Jacobi polynomials. For α∈R/{0}consider the matrix (4.10) R=αA −aelog(q)(J+A)−e−log(q)(J+A), where Aand Jare the matrices given by (1.25). Notice that, since −aelog(q)(J+A)− e−log(q)(J+A)has different eigenvalues, so does R, therefore it can be diagonalized. Let S be a matrix such that S−1RS is diagonal, (4.11) S−1(αA −aelog(q)(J+A)−e−log(q)(J+A))S=−aelog(q)J−e−log(q)J=−aqJ−q−J. We set the matrices (4.12) M=S−1(J+A)S, L =qM=S−1qJ+AS, U =αS−1AS. 4.3. The 2×2case. 59 where ξn=(1+aq∣v∣2(1−qn)(1−bqn) (1−abqn+1)(1−aqn+1))−1 . Proof. Let (Pn)nbe the sequence of orthogonal polynomials given by Pn=Mn˜ Pn. Then, D(Pn))=M−1 n˜ ΛnMnPn=ΛnPn. A straightforward computation gives that for Mnand ˜ Λngiven by (4.31), the following holds Λn=M−1 n˜ ΛnMn=diag(−q−n−1−abqn+2,−q−n−abqn+1). We now write Pn(qx)=Mn˜ Pn(qx)=(˜pn 11(qx)˜pn 12(qx) ˜pn 21(qx)˜pn 22(qx),), Qn(qx)=Pn(qx)Lx=(qx˜pn 11(qx)˜pn 12 −(1−qx)v˜pn 11(qx) qx˜pn 21(qx)˜pn 22 −(1−qx)v˜pn 21(qx))=(rn 11(qx)rn 12(qx) rn 21(qx)rn 22(qx)).(4.35) Taking into account that the sequence of monic orthogonal polynomials, (˜ Pn)nsatisfies (4.30) and that L−xF0(qx)Lxis diagonal (see Theorem 4.1.1) we obtain (DPn)(qx)Lx=Pn(qx−1)(q−1x)L−1Lx+Pn(qx)LxL−x(U−x−1(aL +L−1))Lx +(ax−1−abq)Pn(qx)LLx =(q−x−1)Qn(qx−1)+Qn(qx)q−x(−(q−1+aq)0 0−(1+a)) +(aq−x−abq)Qn(qx+1) =diag(−q−n−1−abqn+2,−q−n−abqn+1)Qn(qx).(4.36) Since the eigenvalues, as well as all the matrix coefficients involved, are diagonal, (4.36) gives four uncoupled scalar-valued q-difference equations. These equations can be matched with the q-difference equation for the little q-Jacobi polynomials, giving pn 11 =κn 11pn(x;aq2, b;q), pn 12 =κn 12pn−1(x;aq2, b;q), rn 21 =κn 21pn+1(x;aq2, b;q), rn 22 =κn 22pn(x;a, b;q), where pn(x;a, b;q)are little q-Jacobi polynomials, and κij ∈Care constants that need to be determined. Then, we can write Pn(x)=(κn 11pn(x;aq2, b;q)κn 12pn+1(x;a, b;q)+κn 11(1−x)vpn(x;aq2, b;q) κn 21pn−1(x;aq2, b;q)κn 22pn(x;a, b;q)+κn 21(1−x)vpn−1(x;aq2, b;q)). From the expression of the leading coefficient of Pn,Mn, the coefficients κn 11 and κn 12 are determined and one can easily see that they are given by (4.34). The expression of Mnalso gives the relation κn 22 =q−n(q, aq;q)n (q−n, abqn+1;q)n−κn 21vqn−1(1−aq)(1−aq2) (1−abqn+1)(1−aqn+1).(4.37) By using the orthogonality of Pnwe can determine completely κn 21 and κn 22 and they are given by (4.34). This completes the proof of the theorem. 60 CHAPTER 4. MATRIX q-POLYNOMIALS. Corollary 4.3.3. For the matrix-valued polynomials (Pn)n≥0with diagonal eigenvalues we have ⟨Pm, Pn⟩W=Hnδm,n, where Hnis the diagonal matrix Hn=diag(∣κn 11∣2hn(aq2, b;q)+∣κn 12∣2hn(a, b;q),∣κn 21∣2hn(aq2, b;q)+∣κn 22∣2hn(a, b;q)), (4.38) and hn(a, b;q)=∥pn(a, b;q)∥2. 4.3.2 Expression as a matrix-valued q-hypergeometric function Let Pn,i be the i-th row of the matrix-valued polynomial Pi. Since the matrix polynomials Pnsatisfy D(Pn)=ΛnPnwhere Dis given by (4.27) and Λnare diagonal matrices, this q-difference equation can be written as two decoupled row equations DPn,i(x)=Pn,i(q−1x)F−1(x)+Pn,i(x)F0(x)+Pn,i(qx)F1(x)=λn,iPn,i,(4.39) where i=1,2, λn,1=−q−n−1−abqn+2,λn,2=−q−n−abqn+1and Pn,i are the rows of the matrix polynomials Pn. We rewrite (4.39) by multiplying on the right by xL, where Lis given by (4.26), we get fro i=1,2, Pn,i(q−1x)(1−x)+Pn,i(x)(x(U−λn,iI)L−(I+aL2))+P(qx)((a−abqx)L2)=0. (4.40) Proposition 4.3.4. The vector-valued orthogonal polynomial solution of (4.40) is given by Pn,i(x)=Pn,i(0)2G1[UL −λn,iL, −abqL2 aL2;q, qx].(4.41) Proof. Since σ(aL2)∩q−N/{0}={1, q2}∩q−N/{0}=∅, we apply Theorem 4.2.2 on (4.40) to get (4.41). Notice that the initial values of the terminating series (4.41) can be determined by Theorem 4.3.2, and are given by Pn(0)=(κn 11 κn 12 +κn 11v κn 21 κn 22 +κn 21v). 4.3.3 The three term recurrence relation and the Rodrigues formula In this section we compute the coefficients of the three term recurrence relation for (Pn)n and we show a Rodrigues formula for them. 4.3. The 2×2case. 61 Theorem 4.3.5. The sequence of orthogonal polynomials (Pn)nsatisfies the three term recurrence relation xPn(x)=AnPn+1(x)+BnPn(x)+CnPn−1,(4.42) where the matrices An,Bnand Cnare given by An=MnM−1 n+1Cn=AnHnH−1 n−1, Bn=−AnPn+1(0)(Pn(0))−1−CnPn−1(0)(Pn(0))−1, and Mnand Hnare given by (4.32) and (4.38) respectively. Proof. Consider the recurrence relation xPn(x)=AnPn+1(x)+BnPn(x)+CnPn−1.(4.43) By comparing the leading coefficients of (4.43) we get An=MnM−1 n+1=⎛⎜⎜⎝ 1−qn(1−q)(1−aq)(1+abq2n+3) (abq2n+2;q2)2 v 0 1 ⎞⎟⎟⎠. From (4.43) we can compute Cnas Cn=An⟨Pn, Pn⟩⟨Pn−1, Pn−1⟩−1. Therefore by Corollary 4.3.3 we can write Cn=AnHnH−1 n−1. To find Bnwe first recall that, by Theorem (4.3.2), Pn(0)=(κn 11 κn 12 +κn 11v κn 21 κn 22 +κn 21v) and det(Pn(0))=κn 11κn 22 −κn 21κn 12 >0, by Theorem 4.3.2. If we plug-in x=0 in (4.43) we find Bn=−AnPn+1(0)(Pn(0))−1−CnPn−1(0)(Pn(0))−1. The following theorem gives a Rodrigues formula for (Pn)n. Theorem 4.3.6. Let R(n)be the matrix function given by R(n)=⎛⎜⎝(1−aqn+2)(1−abqn+3)+av2q2(1−qn)(1−bqn+1) 1−abq2n+30 −(1−qn)avq21−aqn+2⎞⎟⎠. Then, the expression ¯ Pn(qx)=q−xDn q⎛⎝axq(n+1)x(bq;q)x (q;q)x−n LxR(n)(Lx)∗⎞⎠W(qx)−1,(4.44) defines a sequence of matrix polynomials orthogonal with respect to (4.28) 62 CHAPTER 4. MATRIX q-POLYNOMIALS. Before we prove Theorem 4.3.6 we formulate a Lemma that will be proved later. Lemma 4.3.7. For 1<k<n, Dn−k q⎛⎝ax+k−1q(n+1)(x+k−1)(bq;q)x+k−1 (q;q)x+k−n−1 T(qx+k−1)R(n)T(qx+k−1)∗⎞⎠Dk q(qxm)∣x=0=0,(4.45) and Dn−k q⎛⎝ax+k−1q(n+1)(x+k−1)(bq;q)x+k−1 (q;q)x+k−n−1 T(qx+k−1)R(n)T(qx+k−1)∗⎞⎠Dk q(qxm)∣x=∞=0. (4.46) We will also make use of the q-Leibniz rule (4.47) Dn q(f(qx)g(qx))=n ∑ k=0[n k]Dn−k qf(qk+x)Dk qg(qx), and the following formula (4.48) Dn q(f(qx))=1 (1−q)nq(n 2)qnx n ∑ j=0(−1)n−j[n j]q(n−j 2)f(qj+x). Proof of Theorem 4.3.6.We need to prove that for all n≥0, ¯ Pnis a matrix-valued polynomial of degree nwith non-singular coefficient and they satisfy an orthogonality relation with respect to (4.28). First step. ¯ Pnis a matrix polynomial of degree nwith non-singular coefficient. Let us write qxW(qx)=ρ(qx)Lx(Lx)∗, where the matrix Lis given by 4.26 and ρ(qx)=qxax(bq;q)x (q;q)x is the weight associated to the scalar little q-Jacobi polynomials with parameters aand b. Taking into account that LxR(n)(Lx)∗is a matrix-valued polynomial of degree 2 in qxand the q-Leibniz rule (4.47), we can write ¯ Pnas ¯ Pn(qx)=Dn q⎛⎝axq(n+1)x(bq;q)x (q;q)x−n LxR(n)(Lx)∗⎞⎠W(qx)−1 (4.49) =Dn q⎛⎝axq(n+1)x(bq;q)x (q;q)x−n⎞⎠(ρ(x))−1LxR(n)(Lx)∗(Lx(Lx)∗)−1 +[n 1]q Dn−1 q⎛⎝ax+1q(n+1)(x+1)(bq;q)x+1 (q;q)x−n+1⎞⎠(ρ(x))−1Dq(LxR(n)(Lx)∗)(Lx(Lx)∗)−1 +[n 2]q Dn−2 q⎛⎝ax+2q(n+1)(x+2)(bq;q)x+2 (q;q)x−n+2⎞⎠(ρ(x))−1D2 q(LxR(n)(Lx)∗)(Lx(Lx)∗)−1. 4.3. The 2×2case. 63 Let us first deal with the scalar part of (4.49). By making use of the formula (4.48) we can write Dn q⎛⎝axq(n+1)x(bq;q)x (q;q)x−n⎞⎠ρ(qx)−1=1 (1−q)nq(n 2) n ∑ j=0(−1)n−j[n j]q q(n−j 2)q(n+1)jaj ×(bqx+1;q)j(qx−n+j+1;q)n−j =tn(qx),(4.50) Dn−1 q⎛⎝ax+1q(n+1)(x+1)(bq;q)x+1 (q;q)x+1−n⎞⎠ρ(qx)−1=qx (1−q)n−1q(n−1 2) n−1 ∑ j=0(−1)n−j−1[n−1 j]q q(n−j−1 2) ×q(n+1)(j+1)aj+1(bqx+1;q)j+1(qx−n+j+2;q)n−j−1 =qxrn(qx)(4.51) Dn−2 q⎛⎝ax+2q(n+1)(x+2)(bq;q)x+2 (q;q)x+2−n⎞⎠ρ(qx)−1=q2x (1−q)n−2q(n−2 2) n−2 ∑ j=0(−1)n−j−2[n−2 j]q q(n−j−2 2)aj+2 ×q(n+1)(j+2)(bqx+1;q)j+2(qx−n+j+3;q)n−j−2 =q2xsn(qx),(4.52) where tn,rnand snare polynomials of degree n. Moreover, one can directly see that these polynomials have leading coefficients given by tn n=1 (1−q)nq(n 2) n ∑ j=0(−1)n−j[n j]q q(j+1 2)q(n+1)jajbj,(4.53) rn n=1 (1−q)n−1q(n−1 2) n ∑ j=0(−1)n−j−1[n−1 j]q q(j+2 2)q(n+1)(j+1)aj+1bj+1,(4.54) sn n=1 (1−q)n−2q(n−2 2) n ∑ j=0(−1)n−j−2[n−2 j]q q(j+3 2)q(n+1)(j+2)aj+2bj+2.(4.55) By applying the q-Leibniz rule (4.47) to the matrix part of (4.49) we can write Dq(LxR(n)(Lx)∗)(L−x)∗L−x=1 (1−q)qxLxR1(n)L−x,(4.56) D2 q(LxR(n)(Lx)∗)(L−x)∗L−x=1 (1−q)2q2xLxR2(n)L−x, where R1(n)=R(n)−LR(n)L∗, R2(n)=R(n)−(1+q−1)LR(n)L∗+q−1L2R(n)(L∗)2. Taking into account (4.50), (4.51), (4.52) and (4.56), we can write (4.49) as ¯ Pn(qx)=tn(qx)LxR(n)L−x+[n 1]q qxrn(qx) (1−q)qxLxR1(n)L−x+[n 2]q q2xsn(qx) (1−q)2q2xLxR2(n)L−x. (4.57) 64 CHAPTER 4. MATRIX q-POLYNOMIALS. By plugging in the explicit expressions of the polynomials tn,rnand snand of the matrices R(n),R1(n),R2(n)and Lxand carrying direct computations we get that (4.57) is a polynomial of degree nwith non-singular leading coefficient. Second step. The orthogonality relation. To prove that the sequence of polynomials given by (4.44) is orthogonal, we see that for n≥1 and 0 ≤m<n,⟨Pn, xmI⟩W=0 holds. By using the q-Leibniz rule (4.47), the formal identity given in (4.48) Lemma 4.3.7, we get ⟨Pn, xm⟩W=1 q−1 ∞ ∑ x=0 Dn q⎛⎝axq(n+1)x(bq;q)x (q;q)x−n LxR(n)(Lx)∗⎞⎠qxm =Dn−1 q⎛⎝axq(n+1)x(bq;q)x (q;q)x−n LxR(n)(Lx)∗⎞⎠Dq(qxm)∣∞ x=0 +1 q−1 ∞ ∑ x=0 Dn−1 q⎛⎝ax+1q(n+1)(x+1)(bq;q)x+1 (q;q)x+1−n LxR(n)(Lx)∗⎞⎠Dq(qxm) =1 q−1 ∞ ∑ x=0 Dn−1 q⎛⎝ax+1q(n+1)(x+1)(bq;q)x+1 (q;q)x+1−n LxR(n)(Lx)∗⎞⎠Dq(qxm), By repeating this process we obtain ⟨Pn, xm⟩=1 q−1 ∞ ∑ x=0 Dn−m−1 q⎛⎝ax+m+1qn(x+m+2)(bq;q)x+m+1 (q;q)x+m+1−n LxR(n)(Lx)∗⎞⎠Dm+1 q(qxm)qx =0 because Dm+1 q(qxm)=0. This gives the desired result. We now prove Lemma 4.3.7. Proof of Lemma 4.3.7.To see that the first boundary condition (4.45) holds, we use the expression 1 (q;q)x−n=(qx−n+1;q)n (q;q)x , that vanishes at x=0. Any other quantity involved in (4.45) is bounded in x=0, then, we have that (4.45) holds. For (4.46), use that 0 <a<q−1and so ax+k−1q(n+1)(x+k−1)goes to 0 when xgoes to ∞. Since all the other quantities remain bounded when xgoes to ∞, we obtain the desired result. Chapter 5 A family of matrix polynomials satisfying second order differential equations. In this chapter we introduce a family of continuous weight matrices and their sequences of orthogonal polynomials. We start by presenting the weights via an LDU-decomposition (see [75]) and we give an alternative expression of them in terms of scalar Gegenbauer polynomials. We then show that the sequences of orthogonal polynomials are eigenfunctions of two independent differential operators. This property can be exploited to obtain a matrix Pearson equation for the weight matrices in the sense of [12] and a compact Rodrigues formula for the polynomials. We also show how the families of polynomials introduced here can be expressed in terms of matrix hypergeometric functions introduced in Chapter 1, and we give the expression of these matrix polynomials in terms of scalar Racah polynomials and Gegenbauer polynomials. Along the whole chapter we will make use of several summation formulas regarding hypergeometric functions. Such formulas are listed in Section (1.1). For a matter of simplicity in the formulas exhibited in this chapter, we perform a tinny change in the notation used for the matrices, by shifting the index summation range. From now, a matrix M∈C(N+1)×(N+1)will be written as M=N ∑ i=0 N ∑ j=0 MijEi,j, and Ei,j is the matrix with a 1 in its (i+1, j +1)entry, and zero othewise. 5.1 The weight matrix We define a family of weights matrices (W(ν))ν>0absolutely continuous with respect to the Lebesgue measure on (−1,1). Then their densities (that we also denote by W(ν)) are positive semi-definite matrices on (−1,1)so they can be expressed in terms of an LDU decomposition of the form L(x)D(x)L∗(x)where L(x)is a triangular unipotent matrix and D(x)is a positive semi-definite diagonal matrix for all x∈(−1,1). In the rest of the 67 68 CHAPTER 5. CONTINUOUS MATRIX POLYNOMIALS chapter we will use the term weight matrix to refer both to the weight matrix itself and to the matrix functions that defines it, dW (ν)(x)=W(ν)(x)dx. To do so we will use the Gegenabuer polynomials defined by (see, for instance, [76], [59]) (5.1) C(ν) n(x)=(2ν)n n!2F1(−n, n +2ν 1 2+ν;(1−x) 2). The Gegenbauer polynomials satisfy the orthogonality relation (5.2) ∫1 −1C(ν) n(x)C(ν) m(x)(1−x2)ν−1/2dx =(2ν)n√πΓ(ν+1 2) n!(n+ν)Γ(ν)δm,n. For ν>0 and `∈1 2N,N=2`+1, we define the matrix polynomials L(ν)∶[−1,1]→CN×N by (5.3) (L(ν)(x))m,k =⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ 0 if m<k m! (2ν+2k)m−kk!Cν+k m−k(x)if m≥k, for k, m =0,⋯,2`. Notice that they are unipotent lower triangular matrices. We also introduce the diagonal matrix polynomial T(ν)∶[−1,1]→CN×N (T(ν)(x))k,k =t(ν) k(1−x2)k, k =0,⋯,2` t(ν) k=k!(ν)k (ν+1/2)k(2ν+2`)k(2`+ν) (2`−k+1)k(2ν+k−1)k . (5.4) A direct check shows that T(ν)(x)is a positive definite matrix for all x∈(−1,1)and for all ν>0. Definition 5.1.1. Let ν>0and `∈1 2N, set N=2`+1. Let W(ν)be the continuous weight matrix supported on (−1,1)and given by dW (ν)=W(ν)(x)dx, where (5.5) W(ν)(x)=L(ν)(x)T(ν)(x)(L(ν)(x))∗. Observation 5.1.2. Notice that (L(ν)(1))i,j =(i j),(L(ν)(−1))i,j =(−1)i−j(i j),for i≥j, it is also clear that (T(ν)(±1))k,k =t(ν) 0δk,0. If we write W(ν)=(1−x2)ν−1/2W(ν) pol (x)then we get (5.6) (W(ν) pol (1))i,j =t(ν) 0=2`+ν, (W(ν) pol (−1))i,j =(−1)i−j(2`+ν). 5.2. Two differential operators. 75 And the boundary condition is lim x→±1(xB1+B0)W(ν)(x)=0.(5.32) We start by proving the boundary condition (5.32). Because of Observation 5.6 we have that for all i, j =0,⋯2` (lim x→1(xB1+B0)W(ν)(x))i,j =(lim x→1(1−x2)(ν−1/2)(xB1+B0)Wpol(1))i,j =lim x→1(1−x2)(ν−1/2)(x(B1)i,i +(B0)i,i+1+(B0)i,i−1) =lim x→1(1−x2)(ν−1/2)(1−x)(2`+ν)`−i i=0, since ν>0. Similarly, we can see that (lim x→−1(xB1+B0)W(ν)(x))i,j =0,for i, j =0,⋯,2`. For the symmetry equations, we check entry per entry the matrix equations (5.30) and (5.31). We do it for i≤j−1, the other cases can be done similarly. We start with the proof of (5.30). By preforming the matrix product we get that the (i, j)-component of (5.30) is given by (W(ν) pol (x)(xB(ν) 1+B(ν) 0)∗+(xB(ν) 1+B(ν) 0)W(ν) pol (x))i,j =(5.33) −i 2` i−1 ∑ t=0 α(ν) t(i−1, j)C(ν) i+j−2t−1(x)−`−i ` i ∑ t=0 xα(ν) t(i, j)C(ν) i+j−2t(x) +2`−i 2` i+1 ∑ t=0 α(ν) t(i+1, j)C(ν) i+j−2t+1(x)−j 2` i ∑ t=0 α(ν) t(i, j −1)C(ν) i+j−2t−1(x) −`−j ` i ∑ t=0 xα(ν) t(i, j)C(ν) i+j−2t(x)+2`−j 2` i ∑ t=0 α(ν) t(i, j +1)C(ν) i+j−2t+1(x). where we have used the explicit expressions of W(ν)from Theorem 5.1.4 and that xB(ν) 1+ B(ν) 0is tridiagonal. Next, we use the three-recurrence relation for Gegenbauer polynomials, see e.g. [76], [77], xC(ν) r(x)=(r+1) 2(r+ν)C(ν) r+1(x)+(r+2ν−1) 2(r+ν)C(ν) r−1, to get rid of the multiplication by x. Notice that the coefficient of C(ν) m+n+1is equal to α(ν) 0(i, j)i+j+1 2(i+j+ν)j−` `+α(ν) 0(i, j +1)2`−j 2` +α(ν) 0(i, j)i+j+1 2(i+j+ν)i−` `+α(ν) 0(i+1, j)2`−i 2`=0, 76 CHAPTER 5. CONTINUOUS MATRIX POLYNOMIALS what follows from the expression of α(ν) t(i, j)(5.9). Regrouping and using the explicit expressions for α(ν) t(m, n)we get that the right hand side of (5.33) equals i ∑ t=0 α(ν) t(i, j)(i+j−2t+2ν−1) 2`(i+j−2t+ν)C(ν) i+j−2t−1(x) ×[(i+j−2t+ν−1)(i+j+ν−t)(2`−i+1)(i−t) (2`−i−j+t+1)(i+j−2t)(i+ν−t−1) +(i+j−2t+ν−1)(i+j+ν−t)(2`−j+1)(j−t) 2`(2`−i−j+t+1)(i+j−2t)(j+ν−t−1) −(2`−i−j)+(i+j−1−2t+ν)(2`+ν−t)(i+1)(j−t) (i+j−2t)(t+1)(j+ν−t−1) +(i+j−2t+ν−1)(2`+ν−t)(j+1)(i−t) (i+j−2t)(t+1)(i+ν−t−1) −(2`−i−j)(i+j−2t+2ν−2)(i−t)(i+j+ν−t)(j−t)(2`+ν−t) (j+ν−t−1)(i+j−2t)(ν+i−t−1)(2`−i−j+t+1)(t+1)], and it is a straightforward check that the coefficient in square brackets is equal to 0. For the proof of the second symmetry equation we proceed in the same way. For the (i, j)-entry of (5.31) (2ν−1)x((xB(ν) 1+B(ν) 0)W(ν) pol (x))i,j −(1−x2)((xB(ν) 1+B(ν) 0)W(ν) pol (x)′)i,j +(1−x2)[(A(ν) 0)i,i −(A(ν) 0)j,j −(B(ν) 1)i,i](W(ν) pol (x))i,j, we use the three term recurrence for the Gegenbauer polynomials to get rid of the multiplication by xand by x2. We also have to use (5.34) (1−x2)dC(ν) r dx (x)=(r+2ν−1)(r+2ν) 2(r+ν)C(ν) r−1(x)−r(r+1) 2(r+ν)C(ν) r+1(x), see e.g. [76, (4.5.7)] with the convention C(ν) −1(x)=0. So this gives again a sum only involving Gegenbauer polynomials as in the check of the (5.30). Insert the explicit expression of the coefficients α(ν) t(i, j)(5.9) to complete the proof by a straightforward but cubersome computation to see that each coefficient in the expansion of Gegenbauer polynomials vanishes. This completes the proof of the symmetry of E(ν)with respect to W(ν). The symmetry of D(ν)with respect to W(ν).The symmetry equations for D(ν) and W(ν)are 2d dx ((1−x2)W(ν)(x))−(C(ν)−xU(ν))W(ν)(x)=W(ν)(x)(C(ν)−xU(ν))∗, (5.35) d2 dx2((1−x2)W(ν)(x))−d dx ((C(ν)−xU(ν))W(ν)(x))+V(ν)W(ν)(x)=W(ν)(x)(V(ν))∗, (5.36) 5.2. Two differential operators. 77 and the boundary condition are (5.37) lim x→±1(1−x2)W(ν)(x)=0,lim x→±1((1−x2)W(ν)(x))′−(C(ν)−xU(ν))W(ν)(x)=0. We start by proving the symmetry equations. We skip the proof of (5.35) because it is completely similar to that of (5.30). For the symmetry equation (5.36) by making use of (5.35) we see that it suffices to prove (5.38) (W(ν)(x)(C(ν)−xU(ν))∗−(C(ν)−xU(ν))W(ν)(x))′=2(W(ν)(x)V(ν)−V(ν)W(ν)(x)). We prove instead that W(ν)(x)(C(ν)−xU(ν))∗−(C(ν)−xU(ν))W(ν)(x) −2(∫W(ν)(x)dx)V(ν)+2V(ν)(∫W(ν)(x)dx)=0, (5.39) which is obtained by integrating (5.38) with respect to x. Then, (5.38) follows by taking the derivative with respect to x. We prove that the (i, j)component for i<jis zero. The other cases can be proved similarly. By performing the matrix product we see that the (i, j)component of (W(x)(C(ν)−xU(ν))∗−(C(ν)−xU(ν))W(x))is given by ρ(x)i ∑ t=0 C(ν) i+j−2t−1[jα(ν) t(i, j −1)+(2`−j)α(ν) t+1(i, j +1)−iα(ν) t+1(i−1, j)−(2`−i)α(ν) t+1(i+1, j)] (5.40) =−i ∑ t=0 ρ(x)C(ν) i+j−2t−1[(ν−1)(i+j−2t` −2t−2`−1+i)+2(i−t)(j−t)(`+1)] ×(2`−i−j)(i−j)(i+j−1−2t+ν)(i+j+2ν−1−2t) (i+j−2t+ν)(i+j−2t)(i+ν−t−1)(2`−i−j+t+1)(j+ν−t−1). In order to compute the part of (5.39) involving the integrals, we use the following formula for the Gegenbauer polynomials: (5.41) ∫ρ(x)C(ν) r(x)dx =ρ(x)⎛⎝(r+1)C(ν) r+1(x) 2(ν+r)(2ν+r)−(2ν+r−1)C(ν) r−1(x) 2r(ν+r)⎞⎠, which follows from the Rodrigues formulas for the Gegenbauer polynomials [76, (4.5.11)], [76, (4.5.5)] and (5.34). Using (5.41) for the integrals in (5.39) we obtain 2((∫W(ν)(x)dx)V(ν)−V(ν)(∫W(ν)(x)dx))i,j =(i−j)(2`−i−j)ρ(x)i ∑ t=0 α(ν) t(i, j)[−(2ν+i+j−2t−1) (i+j−2t)(ν+i+j−2t) ×α(ν) t(i, j) α(ν) t+1(i, j)(i+j−2t−1 (ν+i+j−2t−2)(2ν+i+j−2t−2)⎤⎥⎥⎥⎥⎦C(ν) i+j−2t−1(x), 78 CHAPTER 5. CONTINUOUS MATRIX POLYNOMIALS which is (5.40). We now prove the boundary condition (5.37). The first one is direct since W(ν)(x)= (1−x2)ν−1/2W(ν) pol (x),ν>0 and W(ν) pol (x)is a matrix polynomial. For the second boundary condition we use the first symmetry equation (5.35) to see that it is equivalent to prove that lim x→±1((W(ν)(x)(xU(ν)−C(ν))∗−(xU(ν)−C(ν))W(ν)(x)))i,j =0,for all i, j =0,⋯,2`. Since U(ν)is a multiple of the identity matrix we get that the second boundary condition for D(ν)holds if and only if limx→±1(W(ν)(x)(C(ν))∗−C(ν)W(ν)(x))i,j =0 for all i, j =0,⋯,2`. By performing the matrix product we get lim x→±1(W(ν)(x)(C(ν))∗−C(ν)W(ν)(x))i,j =lim x→±1((W(ν))i,j−1(x)C(ν) j,j−1+(W(ν))i,j+1(x)C(ν) j,j+1 (5.42) −(C(ν))i,i−1(W(ν)(x))i−1,j −(C(ν))i,i+1(W(ν)(x))i+1,j). From the expression of W(ν)in Definition 5.1.1 it follows that lim x→±1(W(ν)(x))i,j =t(ν) 0lim x→±1(1−x2)ν−1/2L(ν) i,0(x)L(ν) j,0(x). Therefore (5.42) becomes lim x→±1(1−x2)ν−1/2(j L(ν) i,0(x)L(ν) j−1,0(x)+(2`−j)L(ν) i,0(x)L(ν) j+1,0(x) −i L(ν) j,0(x)L(ν) i−1,0(x)−(2`−i)L(ν) j,0(x)L(ν) i+1,0(x)). By introducing the explicit values from the definition of the matrix functions L(ν)(x)and taking into account the expression of the Gegenbauer polynomials (5.1) and the symmetry relation C(ν) n(x)=(−1)nC(ν) n(−x)=(−1)n(2ν)n n!2F1(−n, n +2ν ν+1 2 ;1+x 2), we get that (j L(ν) i,0(x)L(ν) j−1,0(x)+(2`−j)L(ν) i,0(x)L(ν) j+1,0(x) −i L(ν) j,0(x)L(ν) i−1,0(x)−(2`−i)L(ν) j,0(x)L(ν) i+1,0(x)) =(1−x)K1(x)=(1+x)K2(x), where K1(x)and K2(x)are matrix polynomials. Therefore, since ν>0 we have lim x→±1(W(ν)(x)(C(ν))∗−C(ν)W(ν)(x))i,j =0,for all i, j =0,⋯,2`. That is, the second boundary condition in (5.37) holds. So, by Theorem 1.3.7,D(ν)is symmetric with respect to W(ν). 5.2. Two differential operators. 79 It just remains to see that the monic orthogonal polynomials satisfy (5.28) and (5.29). To do so we first notice that E(ν)and D(ν)do not raise the degree of polynomials, so Lemma 1.3.3 applies and we get that the monic orthogonal polynomials are eigenfunctions of D(ν)and E(ν). The expression of the eigenvalues in (5.28) and (5.29) follows directly by equating the corresponding coefficients. For matrix orthogonal polynomials, Cantero, Moral and Vel´azquez [12] established criteria for the derivatives of matrix orthogonal polynomials to be again matrix orthogonal polynomials. A result that is recovered in Theorem 1.3.9. In order to establish the connection with Theorem 1.3.9, we consider the following second order differential operator for ν>01, (5.43) D(ν) (Φ,Ψ)=(E(ν))2+(2`+2)E(ν)+((`+ν)2 `2)D(ν)+ν(ν−1)(2`+ν+1)(ν+2`) `2I. By Theorem 5.2.1,D(ν) (Φ,Ψ)is symmetric with respect to the weight W(ν). Moreover, it commutes with J. Proposition 5.2.1. The matrix differential operator D(ν) (Φ,Ψ)commutes with J. Proof. A straightforward calculation shows that D(ν)commutes with J, and that the differential operator E(ν)satisfies JE(ν)J=−(E(ν)+(`+1)I). Then, by the definition of D(ν) (Φ,Ψ), it is clear that JD(ν) (Φ,Ψ)J=J((E(ν))2+(2`+2)E(ν)+((`+ν)2 `2)D(ν)+ν(ν−1)(2`+ν+1)(ν+2`) `2I)J =(E(ν))2+(2`+2)E(ν)+((`+ν)2 `2)D(ν)+ν(ν−1)(2`+ν+1)(ν+2`) `2I. A straightforward calculation shows that D(ν) (Φ,Ψ)can be written as (5.44) D(ν) (Φ,Ψ)=d2 dx2Φ(ν)(x)∗+d dxΨ(ν)(x)∗, 1The notation of this operator is motivated by equation (5.44) 80 CHAPTER 5. CONTINUOUS MATRIX POLYNOMIALS where Φ(ν)and Ψ(ν)are the explicit matrix polynomials of degree 2 and 1, Φ(ν)(x)=x22` ∑ i=0(`−i)2−(`+ν)2 `2Ei,i +x 2` ∑ i=1(i−1−2`)(2`−2i+1) 2`2Ei,i−1 (5.45) +x 2`−1 ∑ i=0(i+1)(2`−2i−1) 2`2Ei,i+1+2` ∑ i=2(2`−i+2)(2`−i+1) 4`2Ei,i−2 +2` ∑ i=0−i(2`−i+1)−(2`−i)(i+1)+4(`+ν)2 4`2Ei,i +2`−2 ∑ i=0(i+2)(i+1) 4`2Ei,i+2, Ψ(ν)(x)=−x 2` ∑ i=0(2`+2ν+1)(ν+i)(ν+2`−i) `2Ei,i (5.46) −(`+ν+1 2)(2` ∑ i=1(i−1−2`)(ν+i−1) `2Ei,i−1+2`−1 ∑ i=0(i+1)(ν+2`−i−1) `2Ei,i+1). Proposition 5.2.2. Let ν>0. The weight matrix W(ν)(x)and the matrix polynomials Φ(ν)(x)and Ψ(ν)(x)satisfy (W(ν)(x)Φ(ν)(x))′=W(ν)(x)Ψ(ν)(x). Proof. Since D(ν) (Φ,Ψ)is a combination of symmetric differential operators it is symmetric, so it satisfies the symmetry equations (5.47) (Φ(ν)(x)∗W(ν)(x))′′ −(Ψ(ν)(x)∗W(ν)(x))′=0, and the boundary condition (5.48) lim x→−1(Φ(ν)(x)∗W(ν)(x))′−Ψ(ν)(x)∗W(ν)(x)=0 holds. Integrating (5.47) in (−1, x)and taking into account (5.48) we get (Φ(ν)(x)∗W(ν)(x))′=Ψ(ν)(x)∗W(ν)(x). The result follows by taking adjoints. Theorem 5.2.3. Let ν>0,x∈(−1,1)then, W(ν+1)(x)=c(ν)W(ν)(x)Φ(ν)(x), where c(ν)=(2ν+1)(2`+ν+1)`2 ν(2ν+2`+1)(2`+ν)(`+ν). Proof. From Proposition 5.2.2 and taking into account that the boundary condition lim x→−1Φ(ν)(x)∗W(ν)(x)=0 holds, it is equivalent to prove (5.49) c(ν)W(ν)(x)Ψ(ν)(x)=(W(ν+1)(x))′. 5.2. Two differential operators. 81 The proof of this theorem uses the same ingredients as the proof of Theorem 5.2.1. Let us calculate the (i, j)entry of both sides in (5.49). We assume i<j, the other situations can be treated in an analogue way. Now (W(ν)(x)Ψ(ν)(x))i,j can be written explicitly using Theorem 5.1.4 and (5.46). This gives explicit terms involving Gegenbauer polynomials multiplied by x. Then, using the three term recurrence relation for Gegenbauer polynomials and regrouping we can write (W(ν)(x)Ψ(ν)(x))i,j as the following expansion in terms of Gegenbauer polynomials: (5.50) (2ν+1)(2`+ν+1) ν(2`+ν)(`+ν) m ∑ t=0 η(ν) t(i, j)C(ν) i+j−2t+1(x), where η(ν) t(i, j)=−(ν+j)(ν+2`−j)(i+j−2t+1) i+j−2t+να(ν) t(i, j)−(j−2`)(ν+j)α(ν) t(i, j +1) +j(ν+2`−j)α(ν) t(i, j −1)−(ν+j)(ν+2`−j)(2ν+i+j−2t+1) i+j−2t+ν+2α(ν) t−1(i, j), η(ν) 0(i, j)=−(ν+j)(ν+2`−j)(i+j+1) (i+j+ν)α(ν) 0(i, j)−(j−2`)(ν+j)α(ν) 0(i, j +1), η(ν) i+1(i, j)=j(ν+2`−j)α(ν) i(i, j −1)−(ν+j)(ν+2`−j) (j−i+ν)α(ν) i(i, j), for t∈{1,⋯, i}. On the other hand, by (5.34) we can expand (W(ν+1)(x))′ i,j in the same basis leading to (5.51) (1−x2)ν−1/2i ∑ t=0−(i+j−2t+1)(2ν+i+j−2t+1) 2να(ν+1) t(i, j)C(ν) i+j−2t+1(x). To obtain (5.51) we calculate ∫1 −1C(ν) k(x)(W(ν+1) i,j (x))′dx =−∫1 −1 dC(ν) k dx (x)(1−x2)ν+1 2 i ∑ t=0 α(ν+1) t(i, j)C(ν+1) i+j−2t(x)dx. Since dC(ν) k dx (x)=2νC(ν+1) k−1by [76, (4.5.5)], the integral on the right hand side of the previous equation can be calculated using the orthogonality relations for the Gegenbauer polynomials (5.2), leading to (5.51). By a straightforward computation we check that (5.50) and (5.51) are the same, up to the constant, as given in Theorem 5.2.3. By the previous theorems and Theorem 1.3.9 we have the following corollary. Corollary 5.2.4. For ν>0, the sequence (dP (ν) n dx )nis a sequence of matrix orthogonal polynomials with respect to weight function W(ν+1)on [−1,1]. In particular, dP ν n dx (x)= nP(ν+1) n−1(x),n≥1. 82 CHAPTER 5. CONTINUOUS MATRIX POLYNOMIALS Then, we get that d dx connects the sequence (P(ν) n)nwith (P(ν+1) n)nand we say that d dx is a forward shift operator. We now show how to go form (P(ν+1) n)nto (P(ν) n)nvia a differential operator. Lemma 5.2.5. Let ν>0. Define the first order matrix differential operator, T(ν)by (5.52) (T(ν)Q)(x)=dQ dx (x)(Φ(ν)(x))∗+Q(x)(Ψ(ν)(x))∗, then, ⟨dP dx , Q⟩(ν+1)=−c(ν)⟨P, T (ν)Q⟩(ν),for matrix polynomials Pand Q. Proof. For ν>0 and P, Q ∈CN×Nwe have ⟨dP dx , Q⟩(ν+1)=∫1 −1 d dxP(x)W(ν+1)Q(x) =−∫1 −1P(x)(d dx (W(ν+1))Q(x)+W(ν+1)d dxQ(x))dx, where we have used the boundary condition for W(ν+1). Now by applying Lemma 5.2.3 and Proposition 5.2.2 we obtain ⟨dP dx , Q⟩(ν+1)=−c(ν)∫1 −1P(x)(W(ν)Ψ(ν)(x)Q(x)+W(ν)Φ(ν)(x)d dxQ(x))dx =−c(ν)⟨P, T (ν)Q⟩(ν). We can now exploit Lemma 5.2.5 in order to obtain information about the monic matrix orthogonal polynomials (P(ν) n)n. Theorem 5.2.6. Let (P(ν) n)nbe the sequence of monic orthogonal polynomials with respect to W(ν). Then (i) the squared norm of P(ν) n,H(ν) nis a diagonal matrix whose entries are given by (H(ν) n)k,k =√πΓ(ν+1 2) Γ(ν+1)ν(2`+ν+n) ν+n n!(`+1 2+ν)n(2`+ν)n(`+ν)n (2`+ν+1)n(ν+k)n(2`+2ν+n)n(2`+ν−k)n (5.53) ×k!(2`−k)!(n+ν+1)2` (2`)!(n+ν+1)k(n+ν+1)2`−k . (ii) The following Rodrigues formula holds: P(ν) n(x)=G(ν) n dn dxn(W(ν+n)(x))W(ν)(x)−1 (5.54) (G(ν) n)j,k =δj,k (−1)n(ν)n(`+ν+1 2)n(`+ν)n(2`+ν)n (ν+1 2)n(ν+k)n(2`+ν+1)n(2`+2ν+n)n(2`+ν−k)n . 5.2. Two differential operators. 83 Proof. To see that (i) holds we first notice that D(ν) Φ,Ψ=T(ν)○d dx then, T(ν)(P(ν+1) n−1(x))=1 nT(ν)d dxP(ν) n(x)=1 nD(ν) Φ,ΨP(ν) n(x)=1 nK(ν) nP(ν) n(x), where K(ν) ncan be determined from the leading coefficients of Φ(ν)and Ψ(ν). In particular, T(ν)is a backward shift operator. Explicitly, K(ν) nis a diagonal invertible matrix for all ν>0, (K(ν) n)k,k =−(ν+k)(2`+2ν+n) `2(2`+ν−k). Then, we can write H(ν) n=⟨P(ν) n, P (ν) n⟩(ν)=(K(ν) n)−1⟨DΦ,ΨP(ν) n, P (ν) n⟩(ν) =(K(ν) nc(ν) n)−1⟨d dxP(ν) n,d dxP(ν) n⟩(ν+1)=(K(ν) n)−1n2 c(ν) n⟨P(ν+1) n−1, P (ν+1) n−1⟩(ν+1). Iterating, we get H(ν) n=n−1 ∏ i=0(n−i)2(c(ν+i)K(ν+i) n−i)−1H(ν+n) 0, where the order in the product does not matter since all the K(ν+i) n−iare diagonal. By performing the product using the expression (5.21) of H(ν+n) 0, we get the result. For (ii), use Theorem 5.2.3 and (5.49) to see that T(ν)can be written as (5.55) T(ν)Q(x)=(c(ν))−1d dx (Q(x)W(ν+1))(W(ν))−1, x ∈(−1,1), where Q∈CN×N[x]and dgr(T(ν)Q)=dgr(Q)+1. Iterating leads to (n−1 ∏ i=0 c(ν+i))(((QT(ν+n−1))⋯T(ν+1))T(ν))(x)=dn dxn(Q(x)W(ν+n)(x))W(ν)(x)−1. Now take Q(x)=P(ν+n) 0(x)=1, so the previous equality becomes n−1 ∏ i=0 c(ν+i)P(ν) n(x)=(n−1 ∏ i=0 K(ν+i) n−i)−1dn dxn(W(ν+n)(x))W(ν)(x)−1. A direct check shows that Gn=(n−1 ∏ i=0 c(ν+i)K(ν+i) n−i)−1 , and this proves the result. 84 CHAPTER 5. CONTINUOUS MATRIX POLYNOMIALS 5.3 Expression in terms of matrix hypergeometric functions In this section we link the matrix orthogonal polynomials with respect to W(ν)to Tirao’s matrix hypergeometric series 2H1(see defintion 1.40). In order to show this link we switch from the interval [−1,1]to [0,1]using x=1−2u. Set (5.56) R(ν) n(u)=(−1)n2−nP(ν) n(1−2u), Z(ν)(u)=W(ν) pol (1−2u). Hence, the rescaled monic matrix orthogonal polynomials R(ν) nsatisfy (5.57) ∫1 0R(ν) n(u)Z(ν)(u)R(ν) m(u)∗(u(1−u))ν−1/2du =δn,m2−2n−2νH(ν) n. In this setting Theorem 5.2.1 gives the following corollary. Corollary 5.3.1. Let  D(ν)and  E(ν)be the matrix differential operators  D(ν)=u(1−u)d2 du2+d dx( C(ν)−u U(ν))− V(ν), E(ν)=d dx(u B(ν) 1+ B(ν) 0)+ A(ν) 0, where  C(ν), U(ν), V(ν), B(ν) 0, B(ν) 1and  A(ν) 0are given by  C(ν)=−2` ∑ i=0(2`−i) 2Ei,i+1+2` ∑ i=0(2`+2ν+1) 2Ei,i −2` ∑ i=0 i 2Ei,i−1, U(ν)=(2`+2ν+1)I,  V(ν)=−2` ∑ i=0 i(2`−i)Ei,i +(ν−1)(2`+ν+1)I,  B(ν) 0=−2` ∑ i=0(2`−i) 4`Ei,i+1+2` ∑ i=0(`−i) 2`Ei,i +2` ∑ i=0 i 4`Ei,i−1, B(ν) 1=−2` ∑ i=0(`−i) `Ei,i,  A(ν) 0=2` ∑ i=0((`+1)(i−2`) `−(ν−1)(`−i) `)Ei,i. Then,  D(ν)and  E(ν)are symmetric with respect to the weight (u(1−u))ν−1/2Z(ν)(u), and  D(ν)and  E(ν)commute. Moreover, for every integer n≥0,R(ν) n D(ν)=Λn( D(ν))R(ν) n, and R(ν) n E(ν)=Λn( E(ν))R(ν) nwith Λn( D(ν))=((ν−1)(2`+ν+1)−n(2`+2ν+n))I−2` ∑ i=0 i(2`−i)Ei,i, Λn( E(ν))=−(n+ν+2`+1)I−(n+ν+`)2` ∑ i=0 i `Ei,i. We can describe the rows of R(ν) nin terms of matrix hypergeometric functions (see [98]). We first need to study the C2`+1-valued polynomial solutions of P D(ν)=λP,λ∈C. In order to avoid technical problems we consider (5.58) D(ν) α= D(ν)+α E(ν)=u(1−u)d2 du2+d du(C(ν) α−uU(ν) α)−V(ν) α, α ∈R, 5.5. Gegenbauer and Racah polynomials. 91 This is a finite three term recurrence relation for the coefficient c(ν) k,j (n)in the subindex j, which can be solved explicitly in terms of Racah polynomials, (see [102], [77]) c(ν) k,j =c(ν) k,0(−1)j(−2`)j(−n−k)j j!(2ν+2`)j Rk(λ(j);−2`−1,−n−k−ν, ν −1, ν −1) =c(ν) k,0(−1)j(−2`)j(−n−k)j j!(2ν+2`)j 4F3(−j, j +2ν−1,−k, −n−ν−2` ν, −n−k, −2`; 1). (5.79) Switching to the variable x, we find (P(ν) n)k,j =(P(ν) n(x)L(ν)(x))k,j =(−2)nc(ν) k,j (n)(n+k−j)! (2j+2ν)n+k−j C(ν+j) n+k−j(x). Then, the orthogonality relations (5.19) and the orthogonality relations for the (scalarvalued) Gegenbauer polynomials, see (5.2), imply δn,m(Hn)k,i =δn+k,m+i(−2)n+m 2`∧(n+k) ∑ j=0 c(ν) k,j (n)c(ν) i,j (m)t(ν) j(n+k−j)! (2ν+2j)n+k−j√πΓ(ν+j+1/2) (n+k+ν)Γ(ν+j). It follows from (5.79) that (5.80) δn,m(Hn)k,k(−2)−n−m(n+k+ν)Γ(ν)Γ(n+k+2ν) (2`+ν)(n+k)!Γ(ν+1/2)√πΓ(2ν) =∣ck0(n)∣2 2`∧(n+k) ∑ j=0(−2`)j(−n−k)j(2ν−1)j(ν+1/2)j j!(2`+2ν)j(n+k+2ν)j(ν−1/2)j ×Rk(λ(j);−2`−1,−k−n−ν, ν −1, ν −1)Rk+n−m(λ(j);−2`−1,−k−n−ν, ν −1, ν −1), which corresponds to the orthogonality relations for the corresponding Racah polynomials (see [102], [77]). From this we find that the sum in (5.80) equals δn,m M(−2`−n−ν)k(−2`−n−k−2ν+1)k(−2`−ν+1)k(−n−k−ν+1)kk! (−2`−n−k−ν+1)2k(−2`)k(−n−k)k(ν)k , where M=(n+k+ν)2`(2ν)2` (n+k+2ν)2`(ν)2`=(2`+ν)n+k(2ν)n+k (2`+2ν)n+k(ν)n+k . Hence, ∣c(ν) k,0(n)∣2=4−2n(ν)2 n(2`+2ν)2 n (k+ν)2 n(2`+ν−k)2 n . It just remains to see the sign of c(ν) k,0(n). If we take the (k, 0)-entry of the three term recurrence relation for R(ν) n(u)we obtain a polynomial identity in u. If we take the leading coefficient it gives c(ν) k,0(n+1)=−c(ν) k,0(n)(n+k+2ν) 4(n+k+ν)+c(ν) k+1,0(n)(2`−k)(2`−k+ν−1)(n+k+ν) 4(2`−k+n+ν−1)(2`+n−k+ν). 92 CHAPTER 5. CONTINUOUS MATRIX POLYNOMIALS If we plug c(ν) k,0(n)=sign(c(ν) k,0(n))∣c(ν) k,0(n)∣in the equation above, we obtain (ν+n)(2`+2ν+n)sign(c(ν) k,0(n+1)) =−(n+k+2ν)(2`+n+ν−k)sign(c(ν) k,0(n))+(k+ν)(2`−k)sign(c(ν) k+1,0(n)). It then follows that sign(c(ν) k,0(n))=sign(c(ν) k+1,0(n))=−sign(c(ν) k,0(n+1)). This implies that sign(c(ν) k,0(n))=(−1)n. The previous discussion is summarized in the following theorem. Theorem 5.5.2. The polynomials R(ν) nare given by (R(ν) n(u))k,j =c(ν) k,0(n)(−1)j(−2`)j(−n−k)j j!(2ν+2`)j ×4F3(−j, j +2ν−1,−k, −n−ν−2` ν, −n−k, −2`; 1)2F1(j−k−n, n +k+j+2ν j+1 2+ν;u). where the constant c(ν) k,0(n)is given by c(ν) k,0(n)=(−1)n4−n(ν)n(2`+2ν)n (k+ν)n(2`+ν−k)n . Note that Theorem 5.5.2 extends [79, Thm. 6.2]. Switching back to P(ν) nand using the inverse of L(ν)(x)in (5.7) due Cagliero and Koornwinder [11], we get the following corollary. Corollary 5.5.3. Using the notation of Theorem 5.5.2 the monic matrix polynomials have the explicit expansion (P(ν) n(x))k,i =(−2)n i!c(ν) k,0(n)2` ∑ j=i(n+k−j)!(−2`)j(−1)j(−n−k)j (2ν+2j)n+k−j(2ν+j+i−1)j−i(2ν+2`)j (5.81) ×Rk(λ(j);−2`−1,−n−k−ν, ν −1, ν −1)C(ν+j) n+k−j(x)C(1−ν−j) j−i(x). Observation 5.5.1. It isn’t obvious that the right hand side of (5.81) is a polynomial of degree at most nin case k>i. In particular, the coefficients of xpwith p>nare zero. Notice that for k>ithe leading coefficient of the right hand side is zero, and this gives the following non-trivial identity regarding the Racah polynomials. 2` ∑ j=i(ν+j)n+k−j(−2`)j(−1)j(−n−k)j (2ν+2j)n+k−j(2ν+j+i−1)j−i(2ν+2`)j ×(1−ν−j)j−i (j−i)!Rk(λ(j);−2`−1,−n−k−ν, ν −1, ν −1)=0. Bibliography [1] N. Aldenhoven, E. Koelink, A. M. de los R´ıos. Matrix-Valued Little q-Jacobi Polynomials. Preprint.http://arxiv.org/abs/1308.2540.xvi [2] R. ´ Alvarez-Nodarse, A. J. Dur´an and A. M. de los R´ıos. Orthogonal matrix polynomials satisfying second order difference equations. J. Approx. Theory, 169, (2013), 40–55. xiii [3] R. ´ Alvarez-Nodarse, J. C. Medem. q-classical polynomials and the q-Askey and Nikiforov-Uvarov tableaus. J. Comput. Appl. Math.,135 (2001), 157–196. vii [4] G. E. Andrews and R. Askey. Classical orthogonal polynomials. In Orthogonal polynomials and applications (Bar-le-Duc, 1984), Lecture Notes in Math.,Springer, Berlin (1985) 36–62. vii [5] G. E. Andrews, R. Askey and R. Roy. Special functions, volume 71 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 1999. 2 [6] W. N. Bailey. Generalized hypergeometric series. Cambridge Tracts in Mathematics and Mathematical Physics, No. 32. Stechert-Hafner, Inc., New York, 1964. 1,2 [7] H. Bavinck and R. Koekoek. On a difference equation for generalizations of Charlier polynomials. J. Approx. Theory,81(2) (1995), 195–206. xiv [8] S. B¨ochner. Uber Sturm-Liouvillesche Polynomsysteme. Math. Zeit.,29(1) (1929), 730–736. v [9] J. Borrego, M. Castro and A. J. Dur´an. Orthogonal matrix polynomials satisfying differential equations with recurrence coefficients having non-scalar limits. Integral Transforms Spec. Funct.,23(9) (2012), 685–700. viii [10] J. M. Berezans′ki˘ı. Expansions in eigenfunctions of selfadjoint operators. Translated from the Russian by R. Bolstein, J. M. Danskin, J. Rovnyak and L. Shulman. Translations of Mathematical Monographs, Vol. 17. American Mathematical Society, Providence, R.I., 1968. viii [11] L. Cagliero and T.H. Koornwinder. Explicit matrix inverses for lower triangular matrices with entries involving Jacobi polynomials. J. Approx. Theory, (2014) In press. 69,92 93 94 BIBLIOGRAPHY [12] M. J. Cantero, L. Moral and L. Vel´azquez. Matrix orthogonal polynomials whose derivatives are also orthogonal. J. Approx. Theory,146(2) (2007), 174–211. viii, x,xvii,10,14,67,74,79 [13] M. M. Castro and F. A. Gr¨unbaum. Orthogonal matrix polynomials satisfying first order differential equations: a collection of instructive examples. J. Nonlinear Math. Phys.,12 (2005), 63–76. viii [14] M. M. Castro and F. A. Gr¨unbaum. The algebra of differential operators associated to a family of matrix-valued orthogonal polynomials: five instructive examples. Int. Math. Res. Not., pages Art. ID 47602, 33, 2006. viii,x,10,12 [15] M. M. Castro and F. A. Gr¨unbaum. The noncommutative bispectral problem for operators of order one. Constr. Approx.,27 (3) (2008), 329–347. viii [16] T. S. Chihara. An introduction to orthogonal polynomials. Gordon and Breach Science Publishers, New York-London-Paris, 1978. Mathematics and its Applications, Vol. 13.9 [17] A. Conflitti and M. J. Schlosser. Noncommutative hypergeometric and basic hypergeometric equations. J. Nonlinear Math. Phys.,17(4) (2010), 429–443. 56 [18] R. S. Costas, F. Marcell´an. q-Classical orthogonal polynomials: A general difference calculus approach. Acta Appl. Math. 111(1) (2010), 107–128. 10 [19] D. Damanik, A. Pushnitski and B. Simon. The analytic theory of matrix orthogonal polynomials. Surv. Approx. Theory,4(2008), 1–85. viii,6 [20] E. Daneri, A. J. Dur´an. Ratio asymptotics for orthogonal matrix polynomials with unbounded recurrence coefficients. J. Approx. Theory 110(1) (2001), 1–17. viii [21] E. Daneri, A. J. Dur´an. Weak convergence for orthogonal matrix polynomials. Indag. Math. (N.S.) 13(1) (2002), 47–62. viii [22] E. Defez, A. J. Dur´an. Orthogonal matrix polynomials and quadrature formulas. Linear Algebra Appl. 345 (2002), 71–84. viii [23] P. Delsarte, Y. V. Genin and Y. G Kamp. Orthogonal polynomial matrices on the unit circle. IEEE Trans. Circuits and Systems,CAS-25(3) (1978), 149–160. 4 [24] J. J. Duistermaat and F. A. Gr¨unbaum. Differential equations in the spectral parameter. Comm. Math. Phys.,103(2) (1986), 177–240. 12 [25] A. J. Dur´an. A generalization of Favard’s theorem for polynomials satisfying a recurrence relation. J. Approx. Theory,74(1) (1993), 83–109. viii [26] A. J. Dur´an. On orthogonal polynomials with respect to a positive definite matrix of measures. Canad. J. Math.,47 (1) (1995), 1180–1195. viii,6 Bibliography 95 [27] A. J. Dur´an. Markov’s theorem for orthogonal matrix polynomials. Canad. J. Math.,48(6) (1996), 1180–1195. viii [28] A. J. Dur´an. Matrix inner product having a matrix symmetric second order differential operator. Rocky Mountain J. Math.,27(2) (1997), 585–600. ix,10, 11 [29] A. J. Dur´an. Ratio asymptotics for orthogonal matrix polynomials. J. Approx. Theory,100(2) (1999), 304–344. viii [30] A. J. Dur´an. A method to find weight matrices having symmetric second-order differential operators with matrix leading coefficient. Constr. Approx.,29(2) (2009), 181–205. viii,ix,x,10 [31] A. J. Dur´an. Generating orthogonal matrix polynomials satisfying second order differential equations from a trio of triangular matrices. J. Approx. Theory.,161 (1) (2009), 88–113. viii [32] A. J. Dur´an. Rodrigues’ formulas for orthogonal matrix polynomials satisfying second-order differential equations. Int. Math. Res. Not. IMRN,5(2010), 824– 855. viii [33] A. J. Dur´an. Rodrigues’s formulas for orthogonal matrix polynomials satisfying higher-order differential equations. Exp. Math.,20 (1) (2011), 15–24. viii [34] A. J. Dur´an. A miraculously commuting family of orthogonal matrix polynomials satisfying second order differential equations. J. Approx. Theory,163 (12) (2011), 1815–1833. viii,x [35] A. J. Dur´an. The algebra of difference operators associated to a family of orthogonal polynomials. J. Approx. Theory,164(5) (2012),586–610. x,xi,10,17, 18,19 [36] A. J. Dur´an and M. D. de la Iglesia. Second-order differential operators having several families of orthogonal matrix polynomials as eigenfunctions. Int. Math. Res. Not. IMRN, pages Art. ID rnn 084, 24, 2008. viii,x,xiii,10,37,44 [37] A. J. Dur´an and M. D. de la Iglesia. Some examples of orthogonal matrix polynomials satisfying odd order differential equations. J. Approx. Theory,150(2) (2008), 153–174. viii,x [38] A. J. Dur´an and F. A. Gr¨unbaum. Orthogonal matrix polynomials satisfying second-order differential equations. Int. Math. Res. Not.,10 (2004),461–484. viii,ix,8,11,13,14 [39] A. J. Dur´an and F. A. Gr¨unbaum. Structural formulas for orthogonal matrix polynomials satisfying second-order differential equations. I. Constr. Approx., 22(2) (2005), 255–271. viii,x,10 96 BIBLIOGRAPHY [40] A. J. Dur´an and F. A. Gr¨unbaum. A characterization for a class of weight matrices with orthogonal matrix polynomials satisfying second-order differential equations. Int. Math. Res. Not.,23 (2005), 1371–1390. viii [41] A. J. Dur´an and F. A. Gr¨unbaum. Orthogonal matrix polynomials, scalar-type Rodrigues’ formulas and Pearson equations. J. Approx. Theory,134 (2) (2005), 267–280. viii [42] A. J. Dur´an and F. A. Gr¨unbaum. A survey on orthogonal matrix polynomials satisfying second order differential equations. J. Comput. Appl. Math.,178 (1-2) (2005), 169–190. viii,x [43] A. J. Dur´an and F. A. Gr¨unbaum. P. A. M. Dirac meets M. G. Krein: matrix orthogonal polynomials and Dirac’s equation. J. Phys. A,39 (14) (2006), 3655- 3662. viii [44] A. J. Dur´an and F. A. Gr¨unbaum. Matrix orthogonal polynomials satisfying second-order differential equations: coping without help from group representation theory. J. Approx. Theory,148(1) (2007), 35–48. viii [45] A. J. Dur´an and F. A. Gr¨unbaum. Matrix differential equations and scalar polynomials satisfying higher order recursions. J. Math. Anal. Appl.,354(1) (2009), 1–11. viii [46] A. J. Dur´an and F. A. Gr¨unbaum. Orthogonal matrix polynomials, scalar-type Rodrigues’ formulas and Pearson equations. J. Approx. Theory,134(2) (2009), 267–280. x,10 [47] A. J. Dur´an and P. L´opez-Rodr´ıguez. Orthogonal matrix polynomials: zeros and Blumenthal’s theorem. J. Approx. Theory,84(1) (1996), 96–118. viii,6 [48] A. J. Dur´an and P. L´opez-Rodr´ıguez. The Lpspace of a positive definite matrix of measures and density of matrix polynomials in L1.J. Approx. Theory,90 (2), 299–318 viii [49] A. J. Dur´an and P. L´opez-Rodr´ıguez. Density questions for the truncated matrix moment problem. Canad. J. Math.,49 (4) (1997), 708–721 viii [50] A. J. Dur´an and P. L´opez-Rodr´ıguez. N-extremal matrices of measures for an indeterminate matrix moment problem. J. Funct. Anal.,174 (2) (2000), 301–321 viii [51] A. J. Dur´an and P. L´opez-Rodr´ıguez. Orthogonal matrix polynomials. In Laredo Lectures on Orthogonal polynomials and Special Functions, Adv. Theory Spec. Funct. Orthogonal polynomials, Nova Sci. Publ., Hauppauge, NY (2004) 13–44. viii [52] A. J. Dur´an and P. L´opez-Rodr´ıguez. Structural formulas for orthogonal matrix polynomials satisfying second-order differential equations. II. Constr. Approx., 26 (1) (2007), 29–47 viii,x Bibliography 97 [53] A. J. Dur´an, P. L´opez-Rodr´ıguez and E. B. Saff. Zero asymptotic behaviour for orthogonal matrix polynomials. J. Anal. Math.,78 (1999), 37–60 viii [54] A. J. Dur´an and B. Polo. Gaussian quadrature formulae for matrix weights. Linear Algebra Appl.,355 (2002), 119–146. viii [55] A. J. Dur´an and B. Polo. Matrix Christoffel functions. Constr. Approx.,20 (2004), 353–376. viii [56] A. J. Dur´an and A.M. de los R´ıos. The convex cone of weight matrices associated to a second-order matrix difference operator. Integral Transforms Spec. Funct., 25 (8) (2014), 663–679. xi,xiv [57] A. J. Dur´an and V. S´anchez-Canales. Orthogonal matrix polynomials whose differences are also orthogonal. J. Approx. Theory,179 (2014), 112–127. x,xi, 10,47 [58] A. J. Dur´an and W. Van Assche. Orthogonal matrix polynomials and higherorder recurrence relations. Linear Algebra Appl.,219 (1995), 261–280. viii [59] A. Erd´elyi, W. Magnus, F. Oberhettinger and F. G. Tricomi. Higher transcendental functions. Vol. II. Robert E. Krieger Publishing Co., Inc., Melbourne, Fla., 1981. Based on notes left by Harry Bateman, Reprint of the 1953 original. 68 [60] R. Gangolli and V. S. Varadarajan. Harmonic analysis of spherical functions on real reductive groups. Results in Mathematics and related Areas. Springer Verlag, Belin 1988. ix [61] F. R. Gantmacher. The theory of matrices. Vols. 1, 2. Translated by K. A. Hirsch. Chelsea Publishing Co., New York, 1959. 43,46 [62] A. G. Garc´ıa, F. Marcell´an and L. Salto. A distributional study of discrete classical orthogonal polynomials. J. Comput. Appl. Math.,57 (1995), 147–162. 23 [63] G. Gasper and M. Rahman. Basic hypergeometric series, volume 96 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, second edition, 2004. With a foreword by Richard Askey. 1,3 [64] J. S. Geronimo. Matrix orthogonal polynomials on the unit circle. J. Math. Phys.,22(7) (1981), 1359–1365. 4 [65] J. S. Geronimo. Scattering theory and matrix orthogonal polynomials on the real line. Circuits Systems Signal Process.,1(3-4) (1982), 471–495. viii [66] F. A. Gr¨unbaum. Matrix valued Jacobi polynomials. Bull. Sci. Math.,127(3) (2003), 207–214. viii,ix [67] F. A. Gr¨unbaum and M. D. de la Iglesia. Matrix-valued orthogonal polynomials related to SU(N+1), their algebras of differential operators and the corresponding curves. Experiment. Math.,16(2) (2007), 189–207. ix 98 BIBLIOGRAPHY [68] F. A. Gr¨unbaum and M. D. de la Iglesia. Matrix valued orthogonal polynomials arising from group representation theory and a family of quasi-birth-and-death processes. SIAM J. Matrix Anal. Appl.,30(2) (2008), 741–761. viii [69] F. A. Gr¨unbaum, P. Iliev. A noncommutative version of the bispectral problem. J. Comput. Appl. Math.,161(2) (2003), 99–118. 12 [70] F. A. Gr¨unbaum, I. Pacharoni and J. Tirao. Matrix valued spherical functions associated to the complex projective plane. J. Funct. Anal.,188(2) (2002), 350–441. viii,ix [71] F. A. Gr¨unbaum, I. Pacharoni and J. Tirao. Matrix valued orthogoonal polynomials of the Jacobi type Indag. Math.,14(3-4) (2003), 353–366. viii,ix [72] F. A- Gr¨unbaum and Juan Tirao. The algebra of differential operators associated to a weight matrix. Integral Equations Operator Theory,58(4) (2007),449–475. viii,x,10,12,13 [73] W. Hahn. ¨ Uber Orthogonalpolynome, die q-Differenzengleichungen gen¨ugen. Math. Nachr.,2(1949), 4–34. vi [74] D. Holger and B. Reuther. Random block matrices and matrix orthogonal polynomials. J. Theoret. Probab.,23(2), (2010), 378–400. viii [75] R. A. Horn and C. R. Johnson. Matrix analysis. Cambridge University Press, Cambridge, second edition, 2013. 8,67 [76] M. E. H. Ismail. Classical and quantum orthogonal polynomials in one variable, volume 98 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 2009. With two chapters by Walter Van Assche, With a foreword by Richard A. Askey, Reprint of the 2005 original. 1,24,68, 70,75,76,77,81 [77] R. Koekoek, P. A. Lesky and R. F. Swarttouw. Hypergeometric orthogonal polynomials and their q-analogues. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2010. With a foreword by Tom H. Koornwinder. vii,1, 23,24,41,75,91 [78] E. Koelink, M. Van Pruijssen and P. Rom´an. Matrix-valued orthogonal polynomials related to (SU(2)×SU(2),diag).Int. Math. Res. Not. IMRN,24 (2012),5673–5730. ix,xvi [79] E. Koelink, M. Van Pruijssen and P. Rom´an. Matrix-valued orthogonal polynomials related to (SU(2)×SU(2),diag), II. Publ. Res. Inst. Math. Sci.,49(2) (2013), 271–312. xvi,85,88,92 [80] E. Koelink, A. M. de los R´ıos and P. Rom´an. Matrix-valued Gegenbauer polynomials. Preprint.http://arxiv.org/abs/1403.2938.xviii [81] M. G. Kre˘ın. Infinite J-matrices and a matrix-moment problem. Doklady Akad. Nauk SSSR (N.S.),69 (1949), 125–128. viii Bibliography 99 [82] M. G. Kre˘ın. The fundamental propositions of the theory of representations of Hermitian operators with deficiency index (m, m).Ukrain. Mat. ˇ Zurnal,1(2) (1949), 3–66. viii [83] O. E. Lancaster. Orthogonal polynomials defined by difference equations. Amer. J. Math. 63 (1941), 185–207 vii,23 [84] P. L´opez-Rodr´ıguez. Riesz’s theorem for orthogonal matrix polynomials. Constr. Approx. 15 (1) (1999), 135–151. viii [85] P. L´opez-Rodr´ıguez. The Nevanlinna parametrization for a matrix moment problem. Math. Scand. 89 (2) (2001), 245–267. viii [86] P. Lesky. ¨ Uber polynomsysteme, die Sturm-Liouvilleschen sifferenzengleichungen gen¨ugen. Math. Zeit. 78 (1962), 439–445. vii [87] F. Marcell´an, M. A. Pi˜nar and H. O. Yakhlef. Relative asymptotics for orthogonal matrix polynomials with convergent recurrence coefficients. J. Approx. Theory.,111 (2001), 1–30. viii [88] F. Marcell´an, M. A. Pi˜nar and H. O. Yakhlef. Perturbations in the Nevai matrix class of orthogonal matrix polynomials. Linear Algebra Appl.,336 (2001), 131– 254. viii [89] F. Marcell´an and I. Rodr´ıguez. A class of matrix orthogonal polynomials on the unit circle. Linear Algebra Appl.,121 (1989), 233–241. 4 [90] J. C. Medem, R. ´ Alvarez-Nodarse, F. Marcell´an. On the q-polynomials: a distributional study. J. Comput. Appl. Math. 135 (2001), 157–196. vii,10 [91] W. Miller. Symmetry Groups and their Applications. Academic Press, New York, (1972). 8 [92] Luisa Miranian. Matrix valued orthogonal polynomials. ProQuest LLC, Ann Arbor, MI, 2005. Thesis (Ph.D.)–University of California, Berkeley. 6 [93] A. F. Nikiforov and U. B. Uvarov. Special Functions of Mathematical Physics. Birkh¨auser Verlag, Basilea 1988. viii,9,10,23,24 [94] I. Pacharoni and P. Rom´an. A sequence of matrix valued orthogonal polynomials associated to spherical functions. Constr. Approx.,28(2) (2008), 127–147. viii, ix,x,10 [95] I. Pacharoni and J. Tirao. Matrix valued orthogonal polynomials arising from the complex projective space. Constr. Approx.,25(2) (2007), 177–192. viii,ix [96] A. Sinap and W. Van Assche. Orthogonal matrix polynomials and applications. In Proceedings of the Sixth International Congress on Computational and Applied Mathematics (Leuven, 1994),66 (1996), 27–52. viii [97] G. Szeg¨o. Orthogonal polynomials. Amer. Math. Soc. Coll. Pub,23 AMS, Providence, Rhode Island, 1974. v 100 BIBLIOGRAPHY [98] J. A. Tirao. The matrix-valued hypergeometric equation. Proc. Natl. Acad. Sci. USA,100(14) (2003),8138–8141. xv,xvii,15,16,56,84,85,88,90 [99] J. A. Tirao. The algebra of differential operators associated to a weight matrix: a first example. In Groups, algebras and applications, Contem. Math, Amer. Math. Soc., Providence, RI. 537 (2011), 291–324. x [100] N. Ja. Vilenkin. Special functions and the theory of group representation. Trans. of Math. Monographs. , 22] AMS, Providence, Rhode Island, 1986. ix [101] L. Vinet and A. Zhedanov. Representations of the Schr¨odinger group and matrix orthogonal polynomials. J. Phys. A,44(35) (2011), 355201, 28. ix [102] J. A. Wilson. Some hypergeometric orthogonal polynomials. SIAM J. Math. Anal.,11(4) (1980), 690–701. 91