A characterization of the classical orthogonal discrete and q-polynomials

A CHARACTERIZATION OF THE CLASSICAL ORTHOGONAL DISCRETE AND q−POLYNOMIALS M. ALFARO AND R. ´ ALVAREZ-NODARSE Abstract. In this paper we present a new characterization for the classical discrete and q−classical (discrete) polynomials (in the Hahn’s sense). Introduction The classical orthogonal polynomials are very interesting mathematical objects that have attracted the attention not only of mathematicians since their appearance at the end of the XVIII century connected with some physical problems. They are used in several branches of mathematical and physical sciences and they have a lot of useful properties: they satisfy a three-term recurrence relation (TTRR), they are the solution of a second order linear differential (or difference) equation, their derivatives (or finite differences) also constitute an orthogonal family, their generating functions can be given explicitly, among others (for a review see e.g. [2, 6, 15, 16] and the recent [3]). Among such properties, a fundamental role is played by the so-called characterization theorems, i.e., such properties that completely define and characterize the classical polynomials. Obviously not every property characterize the classical polynomials and as an example we can use the TTRR. It is well known that, under certain conditions —by the so-called Favard Theorem (for a review see [11])–, the TTRR characterizes the orthogonal polynomials (OP) but there exist families of OP that satisfy a TTRR but not a linear differential equation with polynomial coefficients, or a Rodriguestype formula, etc. For a more complete review on this see e.g. [1, 7, 12, 14] or the more recent work [2]. In this paper we will complete the works [7, 14] proving a new characterization for the classical discrete [7, 10] and the qclassical [8, 14] polynomials. 1. Preliminaries Let Pbe the linear space of polynomial functions in Cwith complex coefficients and P∗be its algebraic dual space, i.e., P∗is the linear space of 2000 Mathematics Subject Classification. 33C45,33D45. Key words and phrases. Classical Orthogonal Polynomials, Discrete Orthogonal Polynomials, q-polynomials, Characterization Theorems. 1 2 M. ALFARO AND R. ´ ALVAREZ-NODARSE all linear functionals u:P→C. In general, we will represent the action of a functional over a polynomial by u,π,u∈P∗,π∈P, and therefore a functional is completely determined by a sequence of complex numbers u,x n=un,n≥0, the so-called moments of the functional. Definition 1.1. Let (Pn)n≥0be a basis sequence of Psuch that degPn=n. We say that (Pn)n≥0is an orthogonal polynomial sequence (OPS), if and only if there exists a functional u∈P∗such that u,P mPn=knδmn,k n=0,n≥0, where δmn is the Kronecker delta. If the leading coefficient of Pnis equal to 1 for all n, i.e., Pn(x)=xn+···, we say that the sequence (Pn)n≥0is a monic orthogonal polynomial sequence (MOPS) and denote it by (Pn)n≥0= mops(u). It is very well known that a such OPS exists if and only if the linear functional uis quasi-definite. Next, we introduce the forward and backward difference operators defined on Pby ∆:P→ P,∆y(x)=y(x+1)−y(x), ∇:P→ P,∇y(x)=y(x)−y(x−1). For the ∆ operator we have the property ∆[f(x)g(x)] = f(x)∆g(x)+g(x+ 1)∆f(x).(1.1) Also we will use the Jackson q-derivative operator Dqon Pdefined by Dq:P→ P,Dqπ=π(qx)−π(x) (q−1)x,|q|=0,1,(1.2) Notice that in this case we have Dq(π(x)ρ(x)) = ρ(x)Dqπ(x)+π(qx)Dqρ(x)=ρ(qx)Dqπ(x)+π(x)Dqρ(x). (1.3) All the above operators are linear and ∆xn=nxn−1+···,Dqxn=[n]qxn−1,n>0,∆1 = Dq1=0, i.e., Dqπ,∆π∈P. Here, and throughout the paper, [n]q,n∈N, denotes the basic q−number ndefined by [n]q:= qn−1 q−1=1+q+···+qn−1,n>0,[0]q:= 0 .(1.4) Definition 1.2. Let u∈P∗and π∈P. We define the action of the ∆−difference operator ∆on P∗by ∆:P∗→P∗,∆u,π=−u,∆π. We define the action of the q−derivatives Dqon P∗by Dq:P∗→P∗, Dqu,π=−u,Dqπ. Definition 1.3. Let u∈P∗and π∈P. We define a polynomial modification of a functional u, the functional πu,byπu,ρ=u,πρ,∀ρ∈P. A CHARACTERIZATION OF THE CLASSICAL ORTHOGONAL DISCRETE AND q−POLYNOMIALS3 From the above definition and the identities (1.1) and (1.3) it follows that ∆(π(x)u)=π(x−1)∆u+∆π(x−1) u,(1.5) Dq(π(x)u)=π(x/q)Dqu+Dqπ(x/q)u,(1.6) for the discrete and the q-case, respectively. Given a basis sequence of polynomials (Bn)n≥0we define the so-called dual basis of (Bn)n≥0as a sequence of linear functionals (bn)n≥0such that bn,B m=δnm,n,m≥0, Furthermore, if (Pn)n≥0is a MOPS associated to the quasi-definite functional u∈P∗, then their corresponding dual basis (pn)⊂P∗is given by pn=k−1 nPnu,k n=u,P2 n,n≥0.(1.7) Definition 1.4. Let u∈P∗be a quasi-definite functional and (Pn)n≥0= mops(u). We say that u(respectively (Pn)n≥0)isa∆-classical functional (respectively a ∆-classical MOPS), if and only if the sequence (∆Pn+1)n≥0 is also orthogonal. We say that u(respectively (Pn)n≥0)isaq−classical functional (respectively a q−classical MOPS), if and only if the sequence (DqPn+1)n≥0is also orthogonal. In the following (Qn)n≥0will denote either the sequence of monic ∆- differences or q−derivatives of (Pn)n≥0, i.e., either Qn=1 n+1 ∆Pn+1,or Qn=1 [n+1]qDqPn+1, respectively, for all n≥0. It is known that Proposition 1.5. [7, 14] Let (Pn)n≥0= mops(u)and (Qn)n≥0be the sequence of monic ∆-differences or q−derivatives. If (Qn)n≥0= mops(v), then v=φuwhere φ∈P,deg φ≤2. In the next Theorem we collect the characterizations already known of the ∆-classical MOPS and the q−classical MOPS, respectively: Theorem 1.6. Let u∈P∗be a quasi-definite functional and (Pn)n≥0= mops(u). The following statements are equivalent [7]: (a) uand (Pn)n≥0are, respectively, a ∆−classical functional and a ∆−classical MOPS. (b) There exist two polynomials φand ψ,deg φ≤2,deg ψ=1, such that ∆(φu)=ψu.(1.8) (c) There exist two polynomials φand ψ,deg φ≤2,deg ψ=1, and λn∈C,λn=0,n≥1and λ0=0, such that φ∆∇Pn+ψ∇Pn+λnPn=0,n=0,1,2, ... . (1.9) (d) (Pn)n≥0satisfies the distributional Rodrigues formula, i.e., there exist a polynomial φ∈P,deg φ≤2and a sequence of complex numbers rn,rn=0,n≥1such that Pnu=rn∆n(φ(n)u),n≥1where φ(n)(x)= n−1  k=0 φ(x+k),(1.10) 4 M. ALFARO AND R. ´ ALVAREZ-NODARSE Whereas for the q-classical polynomials the following statements are equivalent [14]: (i) uand (Pn)n≥0are, respectively, a q−classical functional and a q−classical MOPS. (ii) There exist two polynomials φand ψ,deg φ≤2,deg ψ=1, such that Dq(φu)=ψu.(1.11) (iii) There exist two polynomials φand ψ,deg φ≤2,deg ψ=1, and λn∈C,λn=0,n≥1and λ0=0, such that φDqD1/qPn+ψD1/qPn+λnPn=0,n=0,1,2, ... . (1.12) (iv) (Pn)n≥0satisfies the distributional Rodrigues formula, i.e., there exist a polynomial φ∈P,deg φ≤2and a sequence of complex numbers rn,rn=0,n≥1such that Pnu=rnDn q(φ(n)u),n≥1where φ(n)(x)= n−1  i=0 φ(qix),(1.13) Moreover, if φ(x)=Ax2+Bx +C, ψ(x)=Mx+M1,M=0,(1.14) and uis quasi-definite then we have the regularity condition nA +M=0 for the ∆-case and [n]qA+M=0for the q-case, respectively. Later, we will use the next technical result: Proposition 1.7. Let (Pn)n≥0= mops uand (Qn)n≥0be the sequence of their monic ∆-differences or q−derivatives, respectively. If uis either a ∆−classical functional or a q−classical functional then, ∆(Qnφu)=(M+nA)Pn+1 u,n≥0,(1.15) Dq(Qnφu)=q−n(M+[n]qA)Pn+1 u,n≥0,(1.16) for the discrete and q-cases, respectively, where M,Aare as in (1.14). Proof. Using (1.7), the Lemmas 1.7 and 1.8 in [7] for the discrete case, and Corollary 2.3 in [14] for the q-case, and proposition 1.5 we find ∆(Qnφu)=−(n+1) k n kn+1 Pn+1 u,Dq(Qnφu)=−[n+1] q k n kn+1 Pn+1 u, A CHARACTERIZATION OF THE CLASSICAL ORTHOGONAL DISCRETE AND q−POLYNOMIALS5 respectively, where kn+1 =u,P2 n+1,k n=φu,Q 2 n. Next we compute k n/kn+1. For the discrete case we have k n:= φu,Q 2 n=1 n+1φu,(x+1) n∆Pn+1 =1 n+1φu,∆(xnPn+1)−∆(xn)Pn+1) =−1 n+1{∆(φu),x nPn+1)+φu,∆(xn)Pn+1)} =−1 n+1u,Mx n+1Pn+1)+u, nAxn+1Pn+1) =−M+nA n+1 kn+1, where we use the equation (1.1) in the second equality and (1.8), (1.14) in the forth one. In the same way, but using (1.3) and (1.11) we find for the q-case k n:= φu,Q 2 n=q−n [n+1] qφu,(qx)nDqPn+1=−q−nM+[n]qA [n+1] q kn+1,  2. Main result In this Section, we prove the characterization Theorem in both situations. 2.1. Classical discrete polynomials. Theorem 2.1. Let u∈P∗be a quasi-definite functional and (Pn)n≥0= mops(u). Then, (Pn)n≥0is a ∆-classical MOPS if and only if for every n≥1, Pnu=∆(αn−1φu),(2.1) where αn−1is a polynomial of degree n−1and φis a polynomial of degree less or equal to 2. Proof. ⇐Taking n= 1 we get P1u=∆(α0φu)=α0∆(φu), thus by Theorem 1.6, uis a classical functional with ψ=P1/α0. ⇒If uis classical then, by Theorem 1.6 (d), Pnu=rn∆n(φ(n)u),n≥1 where φ(n)(x)= n−1  k=0 φ(x+k). Therefore, it is enough to show that rn∆n(φ(n)u)=∆(αn−1φu) being αn−1 a(n−1)-degree polynomial. For doing that, let nbe a fixed positive integer, we prove that rn∆n(φ(n)u)=∆ n−k(αkφ(n−k)u),k=0,1,...,n−1,(2.2) 6 M. ALFARO AND R. ´ ALVAREZ-NODARSE where αkis a polynomial of degree k. Obviously the formula is correct for k= 0 by taking α0(x)=rn. Let suppose that it is true for k=0,1,2,...,p, p<n−1 rn∆n(φ(n)u)=∆ n−p(αpφ(n−p)u)=∆ n−(p+1)∆(αpφ(n−p)u). Let show that ∆(αpφ(n−p)u)=αp+1φ(n−p−1)u,deg αp+1 =p+1. For doing that, notice that φ(n−p)(x)=φ(x)φ(n−p−1)(x+ 1), and therefore using (1.5) ∆(αp(x)φ(n−p)(x)u)=∆(αp(x)φ(n−p−1)(x+1)φ(x)u) =αp(x−1)φ(n−p−1)(x)∆(φ(x)u)+∆(αp(x−1)φ(n−p−1)(x))φ(x)u ={αp(x−1)φ(n−p−1)(x)ψ(x)+∆(αp(x−1)φ(n−p−1)(x))φ(x)}u To complete the proof it suffices to show that Λ(x):=αp(x−1)φ(n−p−1)(x)ψ(x)+∆(αp(x−1)φ(n−p−1)(x))φ(x) ={αp(x−1)ψ(x)+φ(x)∆αp(x−1)}φ(n−p−1)(x) +αp(x)φ(x)∆φ(n−p−1)(x) =αp+1(x)φ(n−p−1)(x), being αp+1 a polynomial of degree p+ 1. Using that ∆φ(n−p−1)(x)=φ(x+n−p−1) −φ(x) φ(x)φ(n−p−1)(x) we finally obtain Λ(x)={αp(x−1)ψ(x)+αp(x)φ(x+n−p−1)−αp(x−1)φ(x)}φ(n−p−1)(x). To prove that the above quotient is a polynomial of degree p+1 we substitute (1.14), αp(x)=apxp+···, and equate the coefficients of the xp+1. This gives ap{M+A(2n−p−2)}= 0, due to the regularity condition (see Theorem (1.6)) and the fact that ap= 0 since the polynomial αphas degree equal to p. This prove (2.2). Now putting k=n−1, the result follows.  To conclude this section notice that comparing (2.1) and (1.15) we obtain that αn−1(x)= Qn−1(x) M+A(n−1) =∆Pn(x) n(M+A(n−1)). 2.1.1. Examples. As examples we will take the monic polynomials of Hahn h(α,β) n(x, N), Meixner m(ν,µ) n(x), Kravchuk k(p) n(x, N), and Charlier c(a) n(x), (see [4]). So we have •Hahn case: Since φ(x)=(x+β+ 1)(N−x−1), ψ(x)=−(α+β+ 2)x−(β+ 1)(N−1), thus αn−1(x)= ∆h(α,β) n(x, N) n(n+α+β+1) =h(α+1,β+1) n−1(x, N −1) n+α+β+1 . A CHARACTERIZATION OF THE CLASSICAL ORTHOGONAL DISCRETE AND q−POLYNOMIALS7 •Meixner case: Now φ(x)=µx +µν,ψ(x)=−(1 −µ)x+µν, and then αn−1(x)=∆m(ν,µ) n(x) n(µ−1) =m(ν+1,µ) n−1(x) µ−1. •Kravchuk case: Since φ(x)=−p 1−p(x−N), ψ(x)=−x+Np 1−p,thus αn−1(x)=∆k(p) n(x, N)) n(µ−1) =(p−1)k(p) n−1(x, N −1). •Charlier case: Since φ(x)=a,ψ(x)=−x+a, therefore αn−1(x)=−∆c(a) n(x) n=c(a) n−1(x). 2.2. q-Classical polynomials. Theorem 2.2. Let u∈P∗be a quasi-definite functional and (Pn)n≥0= mops(u). Then, (Pn)n≥0is a q-classical if and only if for every n≥1, Pnu=Dq(αn−1φu),(2.3) where αn−1is a polynomial of degree n−1and φis a polynomial of degree less or equal to 2. Proof. ⇐Taking n= 1 we get P1u=Dq(α0φu)=α0Dq(φu), thus by Theorem 1.6 (iv), uis a classical functional with ψ=P1/α0. ⇒If uis classical then, by Theorem 1.6 then Pnu=rnDn q(φ(n)u),n≥1 where φ(n)(x)= n−1  k=0 φ(qkx). Therefore, it is enough to show that rnDn q(φ(n)u)=Dq(αn−1φu) being αn−1 a(n−1)-degree polynomial. For doing that we prove that rnDn q(φ(n)u)=Dn−k q(αkφ(n−k)u),k=0,1,...,n−1,(2.4) where αkis a polynomial of degree k. Obviously the formula is correct for k= 0 just taking α0(x)=rn. Let suppose that it is true for k=0,1,2,...,p, p<n−1 Dn q(φ(n)u)=Dn−p q(αpφ(n−p)u)=Dn−(p+1) qDq(αpφ(n−p)u). Let show that Dq(αp(x)φ(n−p)(x)u)=αp+1(x)φ(n−p−1)(x)u,deg αp+1 =p+1. 8 M. ALFARO AND R. ´ ALVAREZ-NODARSE For doing that, notice that φ(n−p)(x)=φ(x)φ(n−p−1)(qx), and therefore using (1.6) Dq(αp(x)φ(n−p)(x)u)=Dq(αp(x)φ(n−p−1)(qx)φ(x)u) =αp(x/q)φ(n−p−1)(x)Dq(φ(x)u)+Dq(αp(x/q)φ(n−p−1)(x))φ(x)u ={αp(x/q)φ(n−p−1)(x)ψ(x)+Dq(αp(x/q)φ(n−p−1)(x))φ(x)}u To complete the proof let show that Λ(x):=αp(x/q)φ(n−p−1)(x)ψ(x)+D(αp(x/q)φ(n−p−1)(x))φ(x) ={αp(x/q)ψ(x)+φ(x)Dqαp(x/q)}φ(n−p−1)(x)+αp(x)φ(x)Dqφ(n−p−1)(x) =αp+1(x)φ(n−p−1)(x), where αp+1 is a polynomial of degree p+ 1. Using that Dqφ(n−p−1)(x)=φ(qn−p−1x)−φ(x) (q−1)x φ(n−p−1)(x) φ(x) we finally obtain Λ(x)=αp(x/q)ψ(x)+αp(x)φ(qn−p−1x)−αp(x/q)φ(x) (q−1)xφ(n−p−1)(x). First of all notice that Λ is a polynomial. To prove that the expression on the brackets is a polynomial of degree p+ 1 we substitute (1.14), αp(x)= apxp+···, and equate the coefficients of the xp+1. This leads to the value apq−p{M+A[2n−p−2]q}, that is different from zero because of the regularity condition (see Theorem (1.6)) and the fact that the polynomial αphas degree equal to p, and then ap= 0. This prove (2.4). Now putting k=n−1, the result follows.  Observe that comparing (2.3) and (1.16) we obtain that αn−1(x)= qnQn−1(x) M+A[n−1]q =qnDqPn(x) [n]q(M+A[n−1]q). 2.2.1. Examples. In this case we have 12 families of classical q-polynomials (see [4]). We will take two representatives examples corresponding to the big q-Jacobi polynomials Pn(x, a, b, c;q) and the little q-Jacobi polynomials pn(x;a, b|q). So we have •Big q-Jacobi polynomials: Since φ(x)=aq(x−1)(bx −c), ψ(x)= 1−abq2 (1−q)qx+a(bq−1)+c(aq−1) 1−q, therefore αn−1(x)= qn [n]q Dqpn(x, a, b|q) 1−abq2 (1−q)q+abq[n−1]q =(1 −q)qn+1 1−abq2 (1−q)q+abq[n−1]q Pn−1(qx,qa,qb,qc;q). A CHARACTERIZATION OF THE CLASSICAL ORTHOGONAL DISCRETE AND q−POLYNOMIALS9 •Little q-Jacobi polynomials: In this case φ(x)=ax(bqx−1), ψ(x)= 1 (1−q)q(1 −abq2)x+aq −1, and thus αn−1(x)=(1 −q)qn+1 [n]q Dqpn(x, a, b, c;q) 1−abq2 (1−q)q+abq[n−1]q =qn 1−abq2 (1−q)q+abq[n−1]q pn−1(x, qa, qb|q). The other ten cases can be obtained in an analogous way or by taking appropriate limits (see e.g. [4, 9]). Acknowledgements. This work was supported by Ministerio de Ciencia y Tecnolog´ıa of Spain (MA & RAN), Diputaci´on General de Arag´on under grant E-64 (MA), and Junta de Andaluc´ıa under grant FQM-262 (RAN). References [1] W. A. Al-Salam, Characterization theorems for orthogonal polynomials. In: Orthogonal Polynomials: Theory and Practice. P. Nevai (Ed.) NATO ASI Series C, Vol. 294. Kluwer Acad. Publ., Dordrecht, 1990, 1-24. [2] R. ´ Alvarez-Nodarse, Polinomios hipergem´etricos y q-polinomios. Monograf´ıas del Seminario Matem´atico “Garc´ıa de Galdeano” Vol. 26. 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