A q-extension of the generalized Hermite polynomials with the continuous orthogonality property on R

International Journal of Pure and Applied Mathematics ————————————————————————– Volume 10 No. 3 2004, 331-342 Aq-EXTENSION OF THE GENERALIZED HERMITE POLYNOMIALS WITH THE CONTINUOUS ORTHOGONALITY PROPERTY ON R R. ´ Alvarez-Nodarse1, M.K. Atakishiyeva2, N.M. Atakishiyev3§ 1Departamento de An´alisis Matem´atico Universidad de Sevilla Apartado Postal 1160, E-41080 Sevilla, SPAIN and 1Instituto Carlos I de F´ısica Te´orica y Computacional Universidad de Granada, E-18071 Granada, SPAIN e-mail: r[email protected] 2Facultad de Ciencias UAEM – Universidad Aut´onoma del Estado de Moredos Apartado Postal 396-3, C.P. 62250, Cuernavaca, Morelos, MEXICO e-mail: m[email protected] 3Instituto de Matem´aticas UNAM – Universidad Nacional Aut´onoma de Mexico Apartado Postal 273-3, C.P. 62210, Cuernavaca Santa Fe 45, Col. Maravillas, Morelos, M´ EXICO e-mail: [email protected] Abstract: In this paper we study in detail a q-extension of the generalized Hermite polynomials of Szeg˝o. A continuous orthogonality property on Rwith respect to the positive weight function is established, a q-difference equation and a three-term recurrence relation are derived for this family of q-polynomials. AMS Subject Classification: 26C05, 33D45, 39A13 Key Words: generalized Hermite polynomials, continuous orthogonality, qdifference equation, three-term recurrence relation Received: November 12, 2003 c 2004, Academic Publications Ltd. §Correspondence author 332 R. ´ Alvarez-Nodarse, M.K. Atakishiyeva, N.M. Atakishiyev 1. Introduction The generalized Hermite polynomials were introduced by Szeg˝o [12] as H(µ) 2n(x) := (−1)n22nn!L(µ−1/2) n(x2), H(µ) 2n+1(x) := (−1)n22n+1 n!x L(µ+1/2) n(x2), (1.1) where µ > −1/2, L(α) n(x) are the Laguerre polynomials, L(α) n(z) := (α+ 1)n n!1F1−n α+ 1  z =(α+ 1)n n! n X k=0 (−n)k (α+ 1)k zk k!,(1.2) and (a)n= Γ(a+n)/Γ(a), n= 0,1,2,..., is the shifted factorial. Observe that the zero value of the parameter µin (1.1) corresponds to the ordinary Hermite polynomials Hn(x), i.e., H(0) n(x) = Hn(x). The generalized Hermite polynomials (1.1) are orthogonal with respect to the weight function |x|2µe−x2,x∈R, i.e., Z∞ −∞ H(µ) n(x)H(µ) m(x)|x|2µe−x2dx = 22nhn 2i! Γ n+ 1 2+µ+1 2δnm,(1.3) where [x] denotes the greatest integer not exceeding x. They satisfy a threeterm recurrence relation 2xH(µ) n(x) = H(µ) n+1(x) + 2(n+ 2µ θn)H(µ) n−1(x), n ≥0,(1.4) and a second-order differential equation xd2 dx2+ 2(µ−x2)d dx + 2nx −2µ θnx−1H(µ) n(x) = 0, n ≥0,(1.5) with θn:= n−2[n/2] (see Szeg˝o [12], Chihara [4]). A detailed discussion of other properties of H(µ) n(x) can be found in Markett [9], Rosenblum [11]. Aq-EXTENSION OF THE GENERALIZED... 333 The reason for interest in studying the generalized Hermite polynomials (1.1) is twofold. Pure mathematically they are of interest as an explicit example of the complete orthonormal set in L2 µ(R), the Hilbert space of Lebesgue measurable functions f(x), x∈R, with ||f||µ:= Z∞ −∞ |f|2|x|2µdx1/2 <∞.(1.6) Hence one can build the Bose-like oscillator calculus in terms of these polynomials, which generalizes the well-known calculus, based on the quantummechanical harmonic oscillator in physics (see, for example, Rosenblum [11]). So we try to make one step further by considering a generalization of the classical Hermite polynomials Hn(x) with two additional parameters, µand q. The aim of this paper is to investigate in detail a q-extension of the generalized Hermite polynomials (1.1) with the continuous orthogonality property on R(the case of discrete orthogonality requires a different technique, see, for example, Berg et al [3]). In Section 2 we introduce this family {H(µ) n(x;q)}in terms of the q-Laguerre polynomials and find a relevant q-difference equation for it. In Section 3 the continuous orthogonality property for {H(µ) n(x;q)}with respect to the positive weight function on Ris explicitly formulated. Section 4 is devoted to the derivation of a three-term recurrence relation for this family of q-polynomials. 2. Generalized Hermite Polynomials It is known from Hahn [7], Exton [5], and Moak [10] that the q-Laguerre polynomials L(α) n(x;q) are explicitly given as L(α) n(x;q) := (qα+1;q)n (q;q)n1φ1 q−n qα+1  q, −qn+α+1 x! =1 (q;q)n2φ1 q−n,−x 0 q, qn+α+1!, (2.1) 334 R. ´ Alvarez-Nodarse, M.K. Atakishiyeva, N.M. Atakishiyev where (a;q)0= 1 and (a;q)n=Qn−1 j=0 (1 −aqj), n= 1,2,..., is the q-shifted factorial, and rφpq−n, a2,··· , ar b1, b2,··· , bp q , z = n X k=0 (q−n;q)k(a2;q)k···(ar;q)k (b1;q)k(b2;q)k···(bp;q)k zk (q;q)kh(−1)kqk(k−1)/2ip−r+1 (2.2) is the basic hypergeometric polynomial of degree nin the variable z(throughout this paper, we will employ the standard notations of the q-special functions theory, see Gasper et al [6] or Andrews et al [2]). The q-Laguerre polynomials (2.1) satisfy two kinds of orthogonality relations, an absolutely continuous one and a discrete one. The former orthogonality relation, in which we are interested in the present paper, is given by Z∞ 0 xα Eq(x)L(α) m(x;q)L(α) n(x;q)dx =d−1 n(α)δmn , α > −1,(2.3) where Eq(x) is the Jackson q-exponential function, Eq(z) := ∞ X n=0 qn(n−1)/2 (q;q)n zn= (−z;q)∞,(2.4) and the normalization constant dn(α) is equal to dn(α) = 1 πsin π(α+ 1) qn(q;q)n (qα+1;q)n (q;q)∞ (q−α;q)∞ .(2.5) The q-Laguerre polynomials (2.1) are defined in such a way that in the limit as q→1 they reduce to the ordinary Laguerre polynomials L(α) n(x), i.e., lim q→1L(α) n((1 −q)x;q) = L(α) n(x).(2.6) We can now define, in complete analogy with the relationship (1.1), a qextension of the generalized Hermite polynomials H(µ) n(x) of the form H(µ) 2n(x;q) := (−1)n(q;q)nL(µ−1/2) n(x2;q), H(µ) 2n+1(x;q) := (−1)n(q;q)nx L(µ+1/2) n(x2;q), (2.7) Aq-EXTENSION OF THE GENERALIZED... 335 which are orthogonal on the real line R. Indeed, since lim q→1 (qa;q)n (1 −q)n= (a)n,(2.8) with the aid of (2.6) one readily verifies that lim q→1(1 −q)−n/2H(µ) n(p1−q x;q) = 2−nH(µ) n(x).(2.9) Observe also that the zero value of the parameter µin (2.7) corresponds to polynomials Hn(x;q)≡ H(0) n(x;q). The sequence {Hn(x;q)}can be expressed either in terms of the q-Laguerre polynomials L(α) n(x;q), α=±1/2 (as it obvious from definition (2.7) itself), or through the discrete q-Hermite polynomials ˜ hn(x;q) of type II: Hn(x;q2) = qn(n−1)/2˜ hn(x;q).(2.10) A detailed discussion of the properties of the polynomials Hn(x;q) can be found in our previous paper ´ Alvarez-Nodarse et al [1] on this subject. Aq-difference equation for the introduced polynomials H(µ) n(x;q) is, in fact, an easy consequence of the known q-difference equation qα(1 + x)L(α) n(q x;q) + L(α) n(q−1x;q) = [1 + qα(1 + qnx)] L(α) n(x;q) (2.11) for the q-Laguerre polynomials (see, for example, formula (3.21.6) in Koekoek et al [8]). Indeed, from this q-difference equation and definition (2.7) it follows immediately that qµ−1/2(1 + x2)H(µ) n(q1/2x;q) + H(µ) n(q−1/2x;q) =hq−θn/2+qµ+(θn−1)/2(1 + q[n/2] x2)iH(µ) n(x;q), (2.12) where, as before, θn=n−2[n/2]. Taking into account that the dilations x→ q±1xare represented by the operators q±xd dx , that is, q±xd dx f(x) = f(q±1x), one now readily verifies that the q-difference equation (2.12) coincides with the second-order differential equation (1.5) in the limit as q→1. 336 R. ´ Alvarez-Nodarse, M.K. Atakishiyeva, N.M. Atakishiyev 3. Orthogonality Relation We begin this section with the following theorem: Theorem 1. The sequence of the q-polynomials {H(µ) n(x;q)}, which are defined by the relations (2.7), satisfies the orthogonality relation ∞ Z −∞ H(µ) m(x;q)H(µ) n(x;q)|x|2µdx Eq(x2) =π cos πµ (q1/2−µ;q)∞ (q;q)∞ q−n 2−µθn(q;q)[n 2](qµ+1/2;q)[n+1 2]δmn , (3.1) on the whole real line Rwith respect to the continuous positive weight function w(x) = 1/Eq(x2). Proof. Since the weight function in (3.1) is an even function of the independent variable xand H(µ) n(−x;q) = (−1)nH(µ) n(x;q) by the definition (2.7), the q-polynomials of an even degree H(µ) 2m(x;q) and of an odd degree H(µ) 2n+1(x;q), m, n = 0,1,2,..., are evidently orthogonal to each other. Consequently, it suffices to prove only those cases in (3.1), when degrees of polynomials mand nare either simultaneously even or odd. Let us consider first the former case. From (2.7) and (2.3) it follows that ∞ Z −∞ H(µ) 2m(x;q)H(µ) 2n(x;q)|x|2µdx Eq(x2) = (−1)m+n(q;q)m(q;q)n ∞ Z −∞ L(µ−1/2) m(x2;q)L(µ−1/2) n(x2;q)|x|2µdx Eq(x2) = 2(−1)m+n(q;q)m(q;q)n ∞ Z 0 L(µ−1/2) m(x2;q)L(µ−1/2) n(x2;q)x2µdx Eq(x2) = (−1)m+n(q;q)m(q;q)n ∞ Z 0 L(µ−1/2) m(y;q)L(µ−1/2) n(y;q)yµ−1/2dy Eq(y) = (q;q)2 nd−1 n(µ−1/2) δmn , Aq-EXTENSION OF THE GENERALIZED... 337 where the normalization constant dn(α) is defined in (2.5). Thus ∞ Z −∞ H(µ) 2m(x;q)H(µ) 2n(x;q)|x|2µdx Eq(x2) =π cos πµ (q1/2−µ;q)∞ (q;q)∞ q−n(q;q)n(qµ+1/2;q)nδmn .(3.2) Likewise, one finds that in the latter case ∞ Z −∞ H(µ) 2m+1(x;q)H(µ) 2n+1(x;q)|x|2µdx Eq(x2) = 2(−1)m+n(q;q)m(q;q)n ∞ Z 0 L(µ+1/2) m(x2;q)L(µ+1/2) n(x2;q)x2(µ+1) dx Eq(x2) = (−1)m+n(q;q)m(q;q)n ∞ Z 0 L(µ+1/2) m(y;q)L(µ+1/2) n(y;q)yµ+1/2dy Eq(y) = (q;q)2 nd−1 n(µ+ 1/2) δmn. Consequently, ∞ Z −∞ H(µ) 2m+1(x;q)H(µ) 2n+1(x;q)|x|2µdx Eq(x2) =π cos πµ (q1/2−µ;q)∞ (q;q)∞ q−n−µ−1/2(q;q)n(qµ+1/2;q)n+1 δmn . (3.3) Putting (3.2) and (3.3) together results in the orthogonality relation (3.1). The positivity of Jackson q-exponential function Eq(x2) for x∈Rand q∈(0,1) is obvious from its definition (2.4): for it is represented as an infinite sum of positive terms (or an infinite product of positive factors). This completes the proof. To conclude this section, we note the obvious fact that in the limit as q→1 the (3.1) reduces to the orthogonality relation (1.3) for the generalized Hermite polynomials (1.1). This follows immediately from the limit relations (2.8) and (2.9), upon using the fact that lim q→1Eq((1 −q)z) = ez.(3.4) 338 R. ´ Alvarez-Nodarse, M.K. Atakishiyeva, N.M. Atakishiyev Also, in the event the parameter µis zero, then the (3.1) coincides with the orthogonality relation for the polynomials (2.10), derived in ´ Alvarez-Nodarse et al [1]. 4. Recurrence Relation In this section we derive a three-term recurrence relation for the q-extension of the generalized Hermite polynomials (2.7). Since an arbitrary family of orthogonal polynomials pn(x) satisfies a recurrence relation of the form (see Chihara [4, p.19]) (anx+bn)pn(x) = pn+1(x) + cnpn−1(x), n ≥0,(4.1) one needs to find coefficients an,bn, and cn, which correspond to the case under discussion. Before starting this derivation we note that in what follows it proves convenient to use the following form L(α) n(x;q) = (qα+1;q)n (q;q)n n X k=0 qk(k+α) (qα+1;q)kn kq (−x)k(4.2) of the q-Laguerre polynomials L(α) n(x;q), which comes from the first line in definition (2.1), upon using the relation (q−n;q)k (q;q)k = (−1)kqk(k−1)/2−nk n kq .(4.3) Let us first consider the case when nin (4.1) is even. Then from (2.7) and (4.2) we find that H(µ) 2n+1(x;q) + c2n(q)H(µ) 2n−1(x;q) = (−1)nx(qµ+3/2;q)n n X k=0 qk(k+µ+1/2) (qµ+3/2;q)kn kq (−x2)k + (−1)n−1c2n(q)x(qµ+3/2;q)n−1 n−1 X k=0 qk(k+µ+1/2) (qµ+3/2;q)kn−1 kq (−x2)k. (4.4) The next step is to employ the relation (1 −qα) (qα+1;q)n= (1 −qn+α) (qα;q)n(4.5) Aq-EXTENSION OF THE GENERALIZED... 339 in order to rewrite the quotient (qµ+3/2;q)n/(qµ+3/2;q)kfrom the first term in the right side of (4.4) as (qµ+3/2;q)n (qµ+3/2;q)k =1−qn+µ+1/2 1−qk+µ+1/2 (qµ+1/2;q)n (qµ+1/2;q)k .(4.6) In the second term in the right side of (4.4) one can use the evident relation (qµ+3/2;q)n−1= (qµ+1/2;q)n/(1 −qµ+1/2) and the same formula (4.5) for the factor (qµ+3/2;q)k. We recall also the property of the q-binomial coefficient n−1 kq =1−qn−k 1−qnn kq .(4.7) Putting this all together, we obtain H(µ) 2n+1(x;q) + c2n(q)H(µ) 2n−1(x;q) = (−1)nx(qµ+1/2;q)n× n X k=0 qk(k+µ+1/2) (qµ+1/2;q)kn kq (−x2)k 1−qk+µ+1/21−qn+µ+1/2−c2n(q)1−qn−k 1−qn. (4.8) The right-hand side of (4.8) should match with H(µ) 2n(x;q) = (−1)n(qµ+1/2;q)n n X k=0 qk(k+µ−1/2) (qµ+1/2;q)kn kq (−x2)k,(4.9) multiplied by a2n(q)x+b2n(q). This means that the coefficient c2n(q) can be found from the equation 1−qn+µ+1/2−c2n(q)1−qn−k 1−qn=dn(q)q−k(1 −qk+µ+1/2),(4.10) where dn(q) is some k-independent factor. It is not difficult to verify that the only solution of the equation (4.10) is c2n(q) = 1 −qnand dn(q) = qn. Thus H(µ) 2n+1(x;q) + (1 −qn)H(µ) 2n−1(x;q) = qnxH(µ) 2n(x;q).(4.11) Similarly, in the case of an odd nfrom (4.8) we have H(µ) 2n+2(x;q) + c2n+1(q)H(µ) 2n(x;q) = (−1)n+1 (qµ+1/2;q)n+1 n+1 X k=0 qk(k+µ−1/2) (qµ+1/2;q)kn+ 1 kq (−x2)k + (−1)nc2n+1(q) (qµ+1/2;q)n n X k=0 qk(k+µ−1/2) (qµ+1/2;q)kn kq (−x2)k.(4.12)