On a modification of the Jacobi linear functional asymptotic properties and zeros of the corresponding orthogonal polynomials

On a Modification of the Jacobi Linear Functional: Asymptotic Properties and Zeros of the Corresponding Orthogonal Polynomials JORGE ARVESÚ1, FRANCISCO MARCELLÁN1and RENATO ÁLVAREZ-NODARSE2 1Departamento de Matemáticas, Universidad Carlos III de Madrid, Avda. de la Universidad, 30, 28911, Leganés, Madrid, Spain 2Departamento de Análisis Matemático, Universidad de Sevilla, Apdo. 1160, E-41080, Seville, Spain. email: [email protected]  Abstract. The paper deals with orthogonal polynomials in the case where the orthogonality condition is related to semiclassical functionals. The polynomials that we discuss are a generalization of Jacobi polynomials and Jacobi-type polynomials. More precisely, we study some algebraic properties as well as the asymptotic behaviour of polynomials orthogonal with respect to the linear functional U U=Jα,β +A1δ(x −1)+B1δ(x +1)−A2δ(x −1)−B2δ(x +1), where Jα,β is the Jacobi linear functional, i.e. Jα,β ,p=1 −1p(x)(1−x)α(1+x)βdx, α,β > −1,p∈P, and Pis the linear space of polynomials with complex coefficients. The asymptotic properties are analyzed in (−1,1)(inner asymptotics) and C\[−1,1](outer asymptotics) with respect to the behaviour of Jacobi polynomials. In a second step, we use the above results in order to obtain the location of zeros of such orthogonal polynomials. Notice that the linear functional Uis a generalization of one studied by T. H. Koornwinder when A2=B2=0. From the point of view of rational approximation, the corresponding Markov function is a perturbation of the Jacobi–Markov function by a rational function with two double poles at ±1. The denominators of the [n−1/n]Padé approximants are our orthogonal polynomials. Mathematics Subject Classifications (2000): 33C45, 42C05. Key words: semiclassical orthogonal polynomials, asymptotics, zeros. 1. Introduction In this work we will study a generalization of the Jacobi polynomials introduced in Koornwinder (1984). Such polynomials are orthogonal with respect to the Jacobi measure ‘perturbed’ by the addition of two delta Dirac measures as well as their derivatives at the points x=±1. 1 Such a kind of modification of a positive linear functional appear when an extension of the Gauss–Lobatto quadrature formulas is considered. In fact, in Bernardi and Maday (1991) such quadrature formulas are used in a spectral method for solving a one-dimensional fourth-order differential problem. Here, the boundary conditions are values of the solution and its first derivative in the ends of the interval (−1,1). The aforementioned modifications were firstly studied in Krall (1940) when he considered the polynomial solution of certain fourth-order linear differential equations. There Krall obtained, apart the classical orthogonal polynomials (Hermite, Jacobi, Laguerre and Bessel), three new families of orthogonal polynomials with respect to positive measures with an absolutely continuous part plus some mass points. More precisely, the so-called classical-type orthogonal polynomials appear. Another approach to this subject was presented in Krall (1981). The analysis of the asymptotic properties of polynomials orthogonal with respect to a perturbation of a measure via the addition of mass points was introduced by Nevai (1979). In particular, he proved how the location of the mass points with respect to the support of the measure has an influence in the asymptotic behaviour of perturbed polynomials. The algebraic properties for such polynomials have attracted the interest of many researchers. A general approach when a modification of a linear functional in the linear space of polynomials with real coefficients via the addition of one delta Dirac measure was started by Chihara (1985) in the positive definite case and Marcellán and Maroni (1992) for quasi-definite linear functionals. From the point of view of differential equations, see Marcellán and Ronveaux (1989). For two point masses there exist very few examples in the literature (see Draïdi, 1990; Koekoek, 1990; Koornwinder, 1984; Kwon and Park, 1997) but the difficulties increase as shown in Draïdi and Maroni (1988). Special emphasis was placed on the modifications of classical linear functionals (Hermite, Laguerre, Jacobi and Bessel) within the framework of the so-called semiclassical orthogonal polynomials. Notice that every positive linear functional induces an inner product in a natural way. However, in general, if we consider a linear functional, no inner product can be defined. Nevertheless, the concept of orthogonality with respect to the linear functional has a sense (see Chihara, 1978; Section 2 in Chapter 1). In Koornwinder (1984), the Jacobi case with two masses at points x=±1 was considered. The hypergeometric representation of the resulting polynomials as well as the existence of a second-order differential equation that such polynomials satisfy have been established. Also the particular cases of the Krall-type polynomials (A. M. Krall, 1981; H. L. Krall, 1940) have been obtained from this general case as special cases or limit cases. The Laguerre case was considered in detail in Koekoek and Koekoek (1991), Koekoek (1988, 1990). The perturbation of a linear functional via the addition of the derivatives of a delta Dirac measure was started in Belmehdi and Marcellán (1992). In particular, necessary and sufficient conditions for the existence of a sequence of polynomials 2 orthogonal with respect to such a linear functional are obtained. Furthermore, an extensive study for the new orthogonal polynomials was performed when the initial functional is semiclassical. This problem can be considered as a limit case of two masses located in two close points. The study of such a kind of modifications of a linear functional has known an increasing interest during the past years since their applications in approximation theory (see Gonchar (1975) for the bounded case and López (1989) for the unbounded one). First, in Álvarez-Nodarse and Marcellán (1995, 1996), the perturbation of the Laguerre linear functional when we add the linear functional M0δ(x) +M1δ(x) is analyzed. More precisely they studied the behavior of the polynomials and their zeros as well as the hypergeometric character of them. More recently, Arvesú et al. (1998) analyzed a generalization of the Bessel polynomials, which appears when one perturbs the Bessel linear functional by the addition of the linear functional M0δ(x)+M1δ(x). In particular, the hypergeometric character of these polynomials and the behavior of their zeros were studied. In this paper, we will deal with the Jacobi case. 1.1. SUMMARY AND STRUCTURE OF THE PAPER In spite of the long history (more than one hundred years ago) of studying of orthogonal polynomials (OP) with respect to semiclassical functionals (see Laguerre, 1885) the theory of semiclassical OP does not enjoy the same level of development and completeness as the theory of classical OP, even, considering the recent powerful modern tools developed to work out with them (see Draïdi, 1990; Draïdi and Maroni, 1988; Kwon and Park, 1997; Marcellán and Maroni, 1992; Maroni, 1991). The present paper intends to cover this void, and introduces a constructive approach to the study of a special class of semiclassical OP. Also, we show that the semiclassical Jacobi-type OP (see Section 3 below) inherits (up to some adaptations) many of the remarkable properties which satisfy classical orthogonal polynomials: exact expression for the coefficients of the hypergeometric series, recurrence relations, etc. The plan of the paper is the following. Section 2 summarizes the basic notions and tools to work out with the classical and semiclassical OP, with particular attention to Jacobi polynomials. In Section 3 we give necessary and sufficient conditions in order to guarantee the existence of the semiclassical orthogonal polynomial sequence. A symmetry property in the same sense as in Koornwinder (1984) is considered. Also, the order of the class for the semiclassical linear functional U (see (19) below) is established, together with the corresponding distributional equation. In Section 4 a general formula for the semiclassical Jacobi-type orthogonal polynomials in terms of the classical ones and their first and second derivatives is given. This fact allows us to study the asymptotic properties which are useful to investigate in Section 5 the location, and asymptotic distribution of zeros of such polynomials. Finally, Section 6 is devoted to conclusions and open problems. 3 2. Preliminaries and Notations Here we present the basic notions, definitions, and notations of the paper. Also, we have enclosed some formulas for the Jacobi polynomials which are useful in the analysis of polynomials orthogonal with respect to the linear functional (19), see Section 3 below. Later on, we summarize the basic tools to work out with semiclassical orthogonal polynomials. 2.1. CLASSICAL ORTHOGONAL POLYNOMIALS:JACOBI POLYNOMIALS In this paper we will always be considering monic orthogonal polynomials from the linear space Pof polynomials with complex coefficients. Pnstands for the subset of polynomials of degree not greater than n. Most of the properties which are used to characterize the classical orthogonal polynomials (OP) in a number of ways (see Chihara, 1978; Szeg˝ o, 1975) follow from the fact that the weight functions ρinvolved in the orthogonality condition  Pn(x)xkρ(x)dx=0,0⩽k<n−1,(1) satisfy the Pearson differential equation [σ(x)ρ(x)]+τ(x)ρ(x) =0,(2) with σ, a polynomial of degree at most 2, and τ, a polynomial of degree exactly 1. The position of the singularities of the first-order differential equation (2) leads to four different possibilities of classical weights (see Table I). One aspect in the theory of classical orthogonal polynomials is that the solutions of the differential equation (2) together with the condition for the path of integration  σ(x)ρ(x)xk|=0,k=0,1,..., (3) determine an integral moment functional U,xk= xkρ(x)dx. (4) Table I. Classical weight functions. Name σ(x) ρ(x) Jacobi x(a −x) xβ(a −x)α Laguerre xx αeβx Hermite const eβx2 Bessel x2xαeγ/x 4 Later, we will return to these types of classical moment functionals to study their generalization. Before proceeding to sketch some remarkable properties as well as some formulas of the Jacobi polynomials, let us specify that we will use the Jacobi polynomials, which are monic polynomials of degree northogonal to all lower degree polynomials with respect to the weight function (1−x)α(1+x)βon [−1,1],where α, β > −1, i.e., they are orthogonal with respect to the linear functional Jα,β on the linear space Pof polynomials with real coefficients defined by Jα,β ,P=1 −1 P (x)(1−x)α(1+x)βdx, α,β > −1,P∈P,(5) and we will denote these monic polynomials by Pα,β n(x). Thus, the orthogonality relation is 1 −1 Pα,β n(x)P α,β m(x)(1−x)α(1+x)βdx=δn,mPα,β n2,(6) where Pα,β n2=2α+β+2n+1n!(n +α+1)(n +β+1) (n +α+β+1)(2n+α+β+1)(n +α+β+1)2 n , and (n)kwith k=1,2,... is the Pochhammer symbol or shifted factorial defined by (n)0:= 1,(n) k:= n(n +1)(n +2)···(n +k−1)=(n +k) (n) . The change of variable x→ (2x/a) −1 gives Jacobi polynomials on [0,a]for the weight function ρ(x) =xβ(a −x)αwith singularities at 0 and a(see Table I). Now, we will list several properties of the Jacobi polynomials, most of which can be found in the literature of special functions, see for instance the classical monograph Orthogonal Polynomials by Szeg˝ o (1975, Chapter 5). The Jacobi polynomials Pα,β n(x) are the polynomial solution of the secondorder linear differential equation of hypergeometric type σ(x)y(x) +τ(x)y(x) +λny(x) =0,(7) where σ(x) =(1−x2), β −α−(α +β+2)x, λn=n(n +α+β+1), respectively. The notation y(k)(x) and Dky(x),k∈Nare used along the paper to denote the kth derivative. (y)(k)(x0)indicates that the kth derivative of yis evaluated at the point x0. 5 The Jacobi polynomials verify the differentiation formula DνPα,β n(x) =n! (n −ν)!Pα+ν,β+ν n−ν(x), ν =0,1,..., (8) where n=1,2,.... Furthermore, the following symmetry property holds Pα,β n(x) =(−1)nPβ,α n(−x). (9) Among other properties there is one very simple and useful for computing aims. It is the exact expression, in terms of the coefficients of the polynomials σand τ only, for the coefficients of the so-called structure relation (1−x2)P  n(x) =˜αnPn+1(x) +˜ βnPn(x) +˜γnPn−1(x), n ⩾0,(10) where Pn(x) =Pα,β n(x), ˜αn=−n, ˜ βn=2n(α −β)(n +α+β+1) (2n+α+β)(2n+2+α+β), ˜γn=4n(n +α)(n +β)(n +α+β)(n +α+β+1) (2n+α+β−1)(2n+α+β)2(2n+α+β+1), (11) and the three-term recurrence relation, xPn(x) =Pn+1(x) +βα,β nPn(x) +γα,β nPn−1(x), n ⩾0,(12) where Pn(x) =Pα,β n(x), βα,β n=β2−α2 (2n+α+β)(2n+2+α+β), γα,β n=4n(n +α)(n +β)(n +α+β) (2n+α+β−1)(2n+α+β)2(2n+α+β+1), (13) providing that P−1=0. The Jacobi polynomials have the following representation as hypergeometric series Pα,β n(x) =2n(α +1)n (n +α+β+1)n 2F1−n, n +α+β+1 α+11−x 2,(14) where pFqa1,a 2,...,a p b1,b 2,...,b qx=∞  k=0 (a1)k(a2)k···(ap)k (b1)k(b2)k···(bq)k xk k!. A consequence of this representation is Pα,β n(1)=2n(α +1)n (n +α+β+1)n ,P α,β n(−1)=(−1)n2n(β +1)n (n +α+β+1)n .(15) 6 Throughout this work we will use Kα,β(p,q) n(x, y) = n  m=0 (P α,β m)(p)(x)(P α,β m)(q)(y) Pα,β m2 =∂p+q ∂xp∂yqKα,β(0,0) n(x, y), (16) in order to denote the kernels of the Jacobi polynomials, as well as their derivatives with respect to xand y, respectively. For p=q=0andn=1,2,... the wellknown Christoffel–Darboux formula n−1  m=0 Pα,β m(x)P α,β m(y) Pα,β m2=Pα,β n(x)P α,β n−1(y) −Pα,β n−1(x)P α,β n(y) (x −y)Pα,β n−12,(17) holds. 2.2. SEMICLASSICAL ORTHOGONAL POLYNOMIALS OF CLASS s: CLASSIFICATION The notion of classical orthogonal polynomials associated with classical weights (see Table I) can be generalized in a very natural way by omitting the restriction on the degrees of the polynomials σand τ(they are polynomials of degree at most 2 and exactly 1, respectively) in Equation (2). So, the moment functional (4) defined by (2) and (3) is called semiclassical of class s,being s=min{max{degσ−2,degτ−1},such that D(σU)=τU}. For s=0 one gets the ordinary classical case, and s>0 corresponds to the semiclassical case of class s. Now, we will list some known results concerning semiclassical linear functionals.Let Ube a linear functional on the linear space Pof polynomials with complex coefficients and let S(U)(z) be its Stieltjes function defined by S(U)(z) =− n⩾0 Un zn+1, where Un=U,xn,n⩾0, are the moments of Uand ·,· means the duality bracket. By a convention, we will suppose that U0=1. Let Pbe the algebraic dual space of Pand Dthe linear space generated by {Dnδ}n⩾0,whereDnδmeans the nth derivative of the Dirac delta in the origin. We consider the isomorphism I:D→Pgiven as follows (see Maroni, 1991): For U= n⩾0 Un (−1)n n!Dnδ, I(U)(z) = n⩾0 Unzn. 7 Then, S(U)(z) =−z−1I(U)(z−1). Let introduce pU,q=U,pqfor every polynomial q(z), and define (Up)(z) = n  m=0n  j=m ajUj−mzm,p(z)= n  j=0 ajzj, and (θ0p)(z) =p(z) −p(0) z. Hence, S(pU)(z) =p(z)S(U)(z) +(Uθ0p)(z), for a polynomial p(z). We define the functional x−1Uand the product of two linear functionals in the following way x−1U,p=U,θ 0p,UV,p=U,Vp. Then it is straightforward to prove that (i) x(x−1U)=U. (ii) x−1(xU)=U−U0δ. (iii) x−2(x2U)=x−1(x−1U)=U−U0δ+U1Dδ. DEFINITION 1. A linear functional Uis said to be a quasi-definite or regular (see Chihara, 1978) functional if there exists a sequence of monic orthogonal polynomials (MOPS), {Pn}n⩾0with respect to U, i.e., it satisfies (i) Pn(x) =xn+lower degree terms. (ii) U,P nPm=knδnm,k n= 0,n=0,1,2,.... AMOPS{Pn}n⩾0with respect to a quasi-definite linear functional satisfies the following three-term recurrence relation Pn+2(x) =(x −βn+1)Pn+1(x) −γn+1Pn(x), n ⩾0, P0(x) =1,P 1(x) =x−β0, with γn= 0,n⩾0andγ0=1=U,P2 0. PROPOSITION 1 (Chihara, 1978). A linear functional Uis quasi-definite if and only if det[Ui+j]n i,j=0= 0,for all n⩾0. Other results concerning the algebra of linear functionals are (see (Bouakkaz and Maroni, 1991; Marcellán and Prianes, 1996; Maroni, 1991) for a more comprehensive approach): 8 LEMMA 1.For p, q ∈Pand for U,V∈P, we have (i) x−1(pU)+U,θ 0pδ=p(x−1U), (ii) q(Uθ0p) −Uθ0(q p) =−θ0[(pU)q], (iii) θ0(Up) =U(θ0p), (iv) U(pq) =(pU)q +xq(Uθ0p), (v) p(UV)=(pV)U+x(Vθ0p)U. In terms of the Stieltjes functions, LEMMA 2. For p∈Pand for U,V∈P, we have (i) S(U)(z) =S(DU)(z), (ii) S(UV)(z) =−zS(U)(z)S(V)(z), (iii) S(x−1U)(z) =z−1S(U)(z), (iv) z−1(Uθ0p)(z) =−z−2S(U,θ 0p, δ)(z−1)+(Uθ2 0p)(z). Finally, THEOREM 1. Let Ube a semiclassical linear functional, and define the set  Xσ={˜x∈C:σ(˜x) =0}. Then, the order of the class of Uis given by s=max{degσ−2,degτ−1}, if and only if one of the following statements holds (i) Either ∀˜x∈ Xσone has τ(˜x) −σ(˜x) = 0. (ii) Or if ˜x∈ Xσsatisfies τ(˜x) −σ(˜x) =0then U,˜τ+˜σ = 0,where˜σ(x) and ˜τ(x) are two polynomials, such that σ(x) =(x −˜x)˜σ(x), τ(x)−σ(x) =(x −˜x)˜τ(x). 2.3. CONNECTION WITH APPROXIMATION THEORY The study of the modification of a measure via the addition of the derivatives of a delta Dirac measure is intimately related with approximation theory (see Gonchar (1975) for the bounded case and López (1989) for the unbounded one). In fact, the denominators qn(x) of the main diagonal sequence for Padé approximants of Stieltjes-type meromorphic functions dµ(x) z−x+ m  j=1 Nj  i=0 Ai,j i! (z −cj)i+1,A Nj,j = 0, 9 kernels and masses A1,A2,B1and B2. For both aims one takes derivatives in (44) and evaluate the resulting equation, as well as (44), at x=1andx=−1. This leads us to a linear system of equations K· Pn=Pn,(45) being K:= I4+K4(n), (46) where I4is the identity matrix and K4(n) =(M1|M2|M3|M4)with the following column vectors M1=A1      Kα,β(0,0) n−1(1,1) Kα,β(0,0) n−1(1,−1) Kα,β(0,0) n−1(1,1) Kα,β(0,1) n−1(1,−1)      +A2      Kα,β(0,1) n−1(1,1) Kα,β(0,1) n−1(−1,1) Kα,β(0,0) n−1(1,1) Kα,β(1,1) n−1(1,−1)       , M2=B1      Kα,β(0,0) n−1(1,−1) Kα,β(0,0) n−1(−1,−1) Kα,β(0,1) n−1(−1,1) Kα,β(0,1) n−1(−1,−1)      +B2      Kα,β(0,1) n−1(1,−1) Kα,β(0,1) n−1(−1,−1) Kα,β(1,1) n−1(1,−1) Kα,β(1,1) n−1(−1,−1)       , M3=A2      Kα,β(0,0) n−1(1,1) Kα,β(0,0) n−1(1,−1) Kα,β(0,1) n−1(1,1) Kα,β(0,1) n−1(1,−1)       ,M4=B2      Kα,β(0,0) n−1(1,−1) Kα,β(0,0) n−1(−1,−1) Kα,β(0,1) n−1(−1,1) Kα,β(0,1) n−1(−1,−1)       .  Pnand Pnare the column vectors  Pn=   Pn(1)  Pn(−1) ( Pn)(1) ( Pn)(−1)   ,Pn=   Pα,β n(1) Pα,β n(−1) (P α,β n)(1) (P α,β n)(−1)   , respectively. Then, by the Cramer’s rule, the system (45) has a unique solution if and only if the determinant of Kis different from zero. Moreover, if Kj(Pn)denotes the matrix obtained substituting the jcolumn in Kby Pn. Then,  Pn(1)=detK1(Pn) detK, Pn(−1)=detK2(Pn) detK, ( Pn)(1)=detK3(Pn) detK,(  Pn)(−1)=detK4(Pn) detK. (47) 16 Conversely, assume that detKdoes not vanish for every n⩾0 and define  Pn(x) by means of expressions (44) and (47). Then, { Pk}k⩾0is a monic OPS with respect to U.✷ Observe that detKdepends on n,α,β,A1,B1,A2,andB2. So, the family of manifolds Fn(α, β, A1,B 1,A 2,B 2)=detK=0 determines a set of parameters for which the existence of semiclassical Jacobi-type orthogonal polynomials is not guaranteed. Howto cover this lack? What kind of conditions are needed towork out with semiclassical Jacobi-type orthogonal polynomials? The following corollary helps us to establish a certain existence condition for the nth OP  Pn(x). COROLLARY 2. Let us define c(α) =A2B2 24(α+1)(α +1)(α +2)(α +3)(α +4),(48) and A1,B1,A2and B2be different from zero in the expression (19). Then, for n large enough, the existence of  Pn(x) is always guaranteed. Furthermore, lim n→∞ detK c(α)c(β)n16+4α+4β=1,α,β>−1, where c(β) is obtained from (48) replacing αby β(α→ β). Proof. Substituting the asymptotic formulas (36)–(38) in (46), and doing a cumbersome calculation we find for nlarge enough a rather lengthy expression for detK(any symbolic computer algebra package like Mathematica can help to calculate it). So, we will not write it here, and provide only the order of nin the power series decomposition of K. More precisely, it is detK∼c(α)c(β)n16+4α+4β. Hence, the corollary holds. ✷ The assumption ‘nlarge enough’ guarantees the existence of the semiclassical Jacobi-type orthogonal polynomials for every nonzero value of the masses A1,B1, A2and B2. So, we work out with these kind of polynomials, without any problem, forgetting about a possible ‘pathological’ election of the masses A1,B1,A2and B2. PROPOSITION 4. The following symmetry properties for the semiclassical Jacobitype orthogonal polynomials and their first derivatives hold Pα,β,A1,B1,A2,B2 n(−x) =(−1)nPβ,α,B1,A1,−B2,−A2 n(x), (49) (P α,β,A1,B1,A2,B2 n)(−x) =(−1)n+1(P β,α,B1,A1,−B2,−A2 n)(x). (50) 17 Proof. Using the definition of the functional U(see (19)) the proof is straightforward. ✷ 3.1. ORDER OF THE CLASS AND DIFFERENTIAL DISTRIBUTIONAL EQUATION FOR U Here we will determine the order of the class for the Jacobi-type moment functional (19), as well as the differential distributional equation that such a functional satisfies. Let us rewrite (19) in the form: U=Jα,β +A1δ(x −1)+B1δ(x +1)−A2δ(x −1)−B2δ(x +1). (51) PROPOSITION 5. The moment linear functional Uverifies the differential distributional equation D[(1−x2)3U]=[β−α−(α +β+6)x](1−x2)2U,(52) being Ua semiclassical functional of class s=4. Proof. The product of (1−x2)2by the functional Uleads to (1−x2)2U=(1−x2)2Jα,β .(53) Now, before taking derivatives in (53), it is convenient to remember the distributional equation for the Jacobi moment functional, i.e., D[(1−x2)Jα,β]=[β−α−(α +β+2)x]Jα,β . Thus, one has D[(1−x2)2U] =D[(1−x2)2Jα,β] =−2x(1−x2)Jα,β +(1−x2)D[(1−x2)Jα,β ] =−2x(1−x2)Jα,β +(1−x2)[β−α−(α +β+2)x]Jα,β =(1−x2)[β−α−(α +β+4)x]Jα,β . If we multiply the above expression by (1−x2),then (1−x2)D[(1−x2)2U]=[β−α−(α +β+4)x](1−x2)2Jα,β =[β−α−(α +β+4)x](1−x2)2U, from which (52) holds. To determine the order of the class it is enough to apply Theorem 1. Thus, s=4. ✷ 18 4. Hypergeometric Character PROPOSITION 6. The semiclassical Jacobi-type orthogonal polynomial  Pn(x) can be represented, up to a multiplicative constant, by a generalized hypergeometric series. More precisely,  Pn(x) =γ2n−3(α +3)n−3 (n +α+β+1)n× ×6F5−n, n +α+β+1,β 0+1,β 1+1,β 2+1,β 3+1 α+3,β 0,β 1,β 2,β 3 1−x 2,(54) where γ,β0,β1,β2and β3are constants depending on n,α,β, and the masses A1, B1,A2, and B2. Proof. Let us define the polynomials in k, a1(k) =8An(n +α)(k +a1)(k +a2), a1=α+1,a 2=a1+1, a2(k) =−4Bn(n +α)(k +b1)(k +b2)(k +b3), b1=−n, b2=−b1+a1+β, b3=a2, a3(k) =8nCn(n +α)(n +α+1) (n +b2)(n +b2+1)(k +c1)(k +c2)(k +c3), c1=a2,c 2=b2,c 3=b2+1, a4(k) =2Dn(n +b2−2)(n +b2−1) (n −1)(k +d1)(k +d2)(k +d3), d1=b1,d 2=b1+1,d 3=a2, a5(k) =2nEn(n +α)(k +e1)(k +e2)(k +e3)(k +e4), e1=b1,e 2=d2,e 3=b2,e 4=b2+1, a6(k) =−4nFn(n +α)(n −1)b2 (n +b2)(n +b2+1)(k +f1)(k +f2)(k +f3)(k +f4), f1=b1,f 2=b2,f 3=b2+1,f 4=b2+2, a7(k) =Gn(n +b2−2)(1−n−b2) (n −2)(k +g1)(k +g2)(k +g3)(k +g4), g1=b1,g 2=b1+1,g 3=b1+2,g 4=b2. Substituting (14) in (64) one finds  Pn(x) =2n−3(α +3)n−3 (n +α+β+1)n ∞  k=07  i=1 ai(k)× ×(−n)k(n +α+β+1)k k!(α +3)k1−x 2k .(55) 19 Taking into account that the expression inside the quadratic brackets is a polynomial in kof degree 4, and denoting it by p4(k) =7  i=1 ai(k):= p(4) 4(k) 4!(k +β0)(k +β1)(k +β2)(k +β3), one can write  Pn(x) =p(4) 4(k) 2n−3(α +3)n−3 4!(n +α+β+1)n ∞  k=0(−n)k(n +α+β+1)k (α +3)k× ×(k +β0)(k +β1)(k +β2)(k +β3) k!1−x 2k,(56) being p(4) 4(x)/4!the leading coefficient of p4(k) p(4) 4(k) 4!=2nEn(n +α) −4nFn(n −1)(n +α)(n +a1) (n +b2)(n +b2+1)− −Gn(n +b2−1)(n +b2−2) (n −2).(57) Since (k +βi)=βi(βi+1)k (βi)k , where −βiwith i=0,1,2,3 are the zeros of p4(k) depending on n,α,β,A1,A2, B1,B2, the expression in (56) becomes  Pn(x) =γ(n) 2n−3(α +3)n−3 (n +α+β+1)n ∞  k=0(−n)k(n +α+β+1)k k!(α +3)k× ×(1+β0)k(1+β1)k(1+β2)k(1+β3)k (β0)k(β1)k(β2)k(β3)k1−x 2k,(58) where γ(n):= 8An(n +α)a1a2+4nBn(n +α)a2b2+ +8nCn(n +α)(n +a1)a2b2(b2+1) (n +b2)(n +b2+1)+ +2nDna2+2n(n −1)En(n +α)b2(b2+1)+ +4n(n −1)Fn(n +α)(n +a1)b2(b2+1)(b2+2) (n +b2)(n +b2+1)+ +n(n −1)Gnb2(n +b2−2)(n +b2−1), (59) 20 which is nothing other than the hypergeometric representation (54). ✷ The zeros of p4(k) are in general complex numbers. If for some i=0,1,2,3, the value of βiis a positive integer we need to take the analytic continuation of the hypergeometric series (54). 5. Some Asymptotic Formulas In this section we will study some asymptotic formulas for the semiclassical Jacobitype orthogonal polynomials. More precisely, the relative asymptotics  Pn(x)/P α,β n(x) outside the interval [−1,1] and the difference between the new polynomials and the classical ones inside [−1,1]. For such a purpose will be useful to have an explicit representation of the semiclassical Jacobi-type orthogonal polynomials in terms of the classical ones. To do that we rewrite Equation (44) in the form  Pn(x) =(1+nζn+nηn)P α,β n(x) + +[ζn(1−x) −ηn(1+x) +(β +1)χn+(α +1)ωn]DPα,β n(x) + +[χn(1+x) −ωn(1−x)]D2Pα,β n(x), (60) where ζn=B1−λnB2 2(β +1)Cβ,α,B1,A1,−B2,−A2 n−B2Dβ,α,B1,A1,−B2,−A2 n, ηn=A1+λnA2 2(α +1)Cα,β,A1,B1,A2,B2 n+A2Dα,β,A1,B1,A2,B2 n, (61) (α +1)χn=A2Cα,β,A1,B1,A2,B2 n, (β +1)ωn=B2Cβ,α,B1,A1,−B2,−A2 n, (62) and Cα,β,A1,B1,A2,B2 n=κα,β n Pn(1), Dα,β,A1,B1,A2,B2 n=κα,β n( Pn)(1). (63) Notice that ζn,η n,χ n,andωndepend on n, α, β, A1,B 1,A 2,andB2. Now, using (8)–(12) we can rewrite (60) as follows  Pn(x) =AnPα,β n(x) +n[BnPα+1,β+1 n−1(x) +CnPα+1,β+1 n(x)]+ +n[DnPα+1,β+1 n−2(x) +(n −1)EnPα+2,β+2 n−2(x)]+ +n(n −1)[FnPα+2,β+2 n−1(x) +GnPα+2,β+2 n−3(x)],(64) 21 where Bn=ζn−ηn+Cnβα+1,β+1 n−1+(β +1)χn+(α +1)ωn, An=1−nCn,C n=−(ζn+ηn), Dn=Cnγα+1,β+1 n−1, En=χn−ωn+Fnβα+2,β+2 n−2,F n=χn+ωn,G n=Fnγα+2,β+2 n−2. (65) Remark 1. The semiclassical Jacobi-type orthogonal polynomials satisfy a second-order linear differential equation (SODE). To deduce it one can rewrite the representation formula (60) in terms of the polynomials and their first derivatives, and using the fact that the Jacobi polynomials satisfy a SODE. THEOREM 2. For nlarge enough the semiclassical Jacobi-type OP satisfy the following outer and inner asymptotics, respectively,  Pn(z) Pα,β n(z) =1+2(β +2) n1−z−1 z+1+ +2(α +2) n1−z+1 z−1+o1 n,(66) where z∈C\[−1,1], 2n+α+β[ Pn(x) −Pα,β n(x)] ∼[sin(θ 2)]−α−3 2[cos(θ 2)]−β−3 2 n× ×(α +β+4)sinθcosnθ +1 2(α +β+1)θ −−1 2(α +1 2)π− −2(α +2)cosnθ +1 2(α +β+3)θ −−1 2(α +3 2)π+O1 n2,(67) x∈(−1,1). Proof. The existence of  Pn(x) for nlarge enough is guaranteed for any choice of nonzero masses A1,B1,A2and B2. Now using the symmetry property (9) and the asymptotic formulas (36) and (40), we can compute the asymptotic behavior of the semiclassical Jacobi-type orthogonal polynomials, as well as their first derivatives at the points ±1, i.e.,  Pn(1)∼√π(α +4) 2n−2A2n7 2+α, Pn(−1)∼(−1)n+1√π(β +4) 2n−2B2n7 2+β, ( Pn)(1)∼−√π(α +3) 2n−1A2n3 2+α,(  Pn)(−1)∼(−1)n+1√π(β +3) 2n−1B2n3 2+β. (68) 22 From (40)–(68) we can give the estimates for the constants defined by (61)–(63) Cα,β,A1,B1,A2,B2∼2(α +4) A2(α +1)n4, Dα,β,A1,B1,A2,B2∼− (α +3) A2(α +1)n2, ζn∼2(β +2) n2,η n∼2(α +2) n2,χ n∼2(α +2)(α +3) n4, ωn∼−2(β +2)(β +3) n4. (69) Finally, from (60), taking derivatives twice and using (40) and (69), we find ( Pn)(1)∼−√πnα+9 2(α +2)(α +5) (α +5)2n+α+β+2.(70) To obtain the relative asymptotics  Pn(z)/P α,β n(z), outside the interval [−1,1] we need to do some handling. First, we multiply (60) by σ(z), and using the SODE (7) we find the following equivalent representation formula σ(z) Pn(z) =a(z;n)P α,β n(z) +b(z;n)DP α,β n(z), (71) where a(z;n), b(z;n) are polynomials of uniformly bounded degree in zwith coefficients depending on ngiven by a(z;n) =(1+nζn+nηn)σ (z) −λn[χn(1+z) −ωn(1−z)], b(z;n) =[ζn(1−z) −ηn(1+z) +(β +1)χn+(α +1)ωn]σ(z)− −τ(z)[χn(1+z) −ωn(1−z)]. (72) Second, we will rewrite (71) in the form  Pn(z) =˜a(z;n)P α,β n(z) +˜ b(z;n)DP α,β n(z), (73) where ˜a(z;n) =(1+nζn+nηn)−λnχn (1−z) −ωn (1+z) ∼1+2(α +β+4) n− −2 n2(α +2)(α +3) (1−z) +(β +2)(β +3) (1+z) ,(74) ˜ b(z;n) =[ζn(1−z) −ηn(1+z) +(β +1)χn+(α +1)ωn]− −τ(x)χn (1−z) −ωn (1+z) 23 ∼2 n2[(β +2)(1−z) −(α +2)(1+z)]+ +2 n4[(β +1)(α +2)(α +3)−(α +1)(β +2)(β +3)]− −2[β−α−(α +β+2)z] n4× ×(α +2)(α +3) (1−z) +(β +2)(β +3) (1+z) .(75) Then, from (73) and using (74)–(76), as well as, 1 n DPα,β n(z) Pα,β n(z) =1 √z2−1+o(1), (76) we obtain the following estimate for the ratio (66). In order to obtain the asymptotic behavior of the difference between the new polynomials and the classical ones, when zbelongs to [−1,1], we use the Darboux formula for the asymptotics of the Jacobi polynomials on the interval θ∈ [ε, π −ε],0<ε1(Szeg˝ o, 1975, Equation 8.21.10, p. 196) anPα,β n(cosθ) =(sin θ 2)−α−1 2(cos θ 2)−β−1 2 √nπ × ×cosnθ +1 2(α +β+1)θ −1 2(α +1 2)π+O1 n3 2,(77) with an=(n+α+β+1)n 2nn!∼2n+α+β √nπ . The expression (64), as well as the following asymptotic estimates for the coefficients An∼1+2(α +β+4) n,B n∼2(β −α) n2, Cn∼−2(α +β+4) n2,D n∼−(α +β+4) 2n2, En=O1 n6,F n=O1 n4,G n=O1 n4, (78) follow from (69). Then, using (64) and (77)–(78) we deduce (67). ✷ 6. Zeros Here we will study the properties of the zeros of the semiclassical Jacobi-type orthogonal polynomials, for nonzero values of the masses, and will present some results concerning their asymptotic behaviour. 24 THEOREM 3.Suppose nN(N∈N). Then, the semiclassical Jacobi-type orthogonal polynomial  Pn(x) has at least n−4different real zeros in (−1,1). Proof. Since, for nlarge enough U, Pn(x) (1−x2)2=Jα,β , Pn(x) (1−x2)2=0, the semiclassical orthogonal polynomial  Pn(x) changes its sign in the interval (−1,1).Letx1,x 2,...,x kbe the different real zeros of odd multiplicity of  Pn(x) in (−1,1). Hence, if q(x) =(x −x1)(x −x2)...(x−xk), then the product  Pn(x)q(x) does not change its sign in (−1,1). Now define h(x) =(1−x2)2q(x). Thus U, Pn(x) h(x)=Jα,β , Pn(x) h(x)>0, so degh(x) ⩾n, i.e., k⩾n−4. ✷ COROLLARY 3. Suppose that all the masses involved in the linear functional U (see (19)) are real, and A2,B 2have the same sign. Then, for nlarge enough the semiclassical Jacobi-type orthogonal polynomial has exactly n−3different real zeros belonging to the interval (−1,1). Proof. Let us consider the case of even nand A2,B 2>0 (the procedure to prove the other cases is completely analog to the developed here). Since  Pn(1)>0and P n(1)<0 (see (68)), then for some positive x>1, the polynomial  Pn(x) has a minimum. This implies that on the right of x=1ithas only two zeros (see the argument below for the point −1), which can be complex conjugates, real and simple or with multiplicity 2. Again from (68), since  Pn(−1) and  P n(−1)are negative the polynomial  Pn(x) is a convex upward function for x<−1 and has a simple real zero; otherwise the number of zeros off [−1,1] would be greater than 4, which yields a contradiction. ✷ COROLLARY 4. Let A1,A2,B1,B2be real numbers, and A2>0,B2<0. Then, for nlarge enough the semiclassical Jacobi-type orthogonal polynomial has exactly 4zeros off [−1,1]. Two of them are located on the right of x=1, and the other two are on the left of x=−1.Thus,in(−1,1)are n−4real and simple zeros. Proof. Its enough to take into account the formulas (68) and analyze two cases: First, when nis even; second, when nis odd. 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