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Fourier-Gauss Transforms of Bilinear Generating Functions for the Continuous Q-Hermite Polynomials

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Fourier-Gauss Transforms of Bilinear Generating Functions for the Continuous Q-Hermite Polynomials Fourier-Gauss Transforms of Bilinear Generating Functions for the Continuous q-Hermite Polynomials M. K. Atakishiyeva∗ N. M. Atakishiyev†‡ CRM-2586 January 1999 ∗Facultad de Ciencias, UAEM, Apartado Postal 396-3, C.P. 62250, Cuernavaca, Morelos, Mexico †Instituto de Matematicas, UNAM, Apartado Postal 273-3, C.P. 62210, Cuernavaca, Morelos, Mexico; [email protected] ‡On leave from: Institute of Physics, Azerbaijan Academy of Sciences, H. Javid Prospekt 33, Baku 370143, Azerbaijan Abstract The classical Fourier-Gauss transforms of bilinear generating functions for the continuous q-Hermite polynomials of Rogers are studied in detail. Our approach is essentially based on the fact that the q-Hermite functions have simple behaviour with respect to the Fourier integral transform with the q-independent exponential kernel. R´esum´e Les transform´eesde Fourier-Gauss pour les fonctions bilin´eairesg´en´eratricesdes q- polynˆomesd’Hermite de Rogers sont ´etudi´eesen d´etail.Notre approche est essentiel- lement bas´eesur le fait que les q-fonctions d’Hermite on un comportement simple par rapport `ala transform´eede Fourier int´egraleavec le noyau exponentiel q-ind´ependant. 1 Introduction Bilinear generating functions (or Poisson kernels) are important tools for studying various prop- erties of the corresponding families of orthogonal polynomials. For example, Wiener has used the bilinear generating function for the Hermite polynomials Hn(x) in proving that the Hermite func- 2 tions Hn(x) exp( x /2) are complete in the space L2 over ( , ) and a Fourier transform of any − −∞ ∞ function from L2 belongs to the same space [1]. Also, it turns out that a particular limit value of the Hermite bilinear generating function reproduces the kernel exp(ixy) of Fourier transformation between two L2 spaces. This idea was imployed for finding an explicit form of the reproducing kernel for the Kravchuk and Charlier functions in [2], whereas the case of the continuous q-Hermite functions was considered in [3]. It is clear that an appropriate q-analogue of the Fourier transform will be an essential ingredient of a completely developed theory of q-special functions. But the point is that the classical Fourier transform with the q-independent kernel turns out to be very useful in revealing close relations between some families of orthogonal q-polynomials [4, 5], as well as among various q-extensions z of the exponential function e [6, 7] and of the Bessel function Jν(z)[8, 9]. One of the possible explanations of this remarkable circumstance is the simple Fourier-Gauss transformation property 1 ∞ irs s2/2 1/4 r2/2 Z e − xq(s) ds = q x1/q(r)e− , (1.1) √2π −∞ iκs enjoyed by the q-linear xq(s) = e and, consequently, by the q-quadratic lattices xq(s) = sin κs 2 or xq(s) = cos κs as well, where q = exp( 2κ ). It remains only to remind the reader that there exist a large class of polynomial solutions− to the hypergeometric-type difference equation, which are defined in terms of these nonuniform lattices (see [10] for a review). Convinced of the power of classical Fourier transform, we wish to apply it for studying some additional properties of bilinear generating functions for the continuous q-Hermite polynomials of Rogers. 2 Linear generating functions The continuous q-Hermite polynomials Hn(x q), q < 1, introduced by Rogers [11], are defined by their Fourier expansion | | | n n i(n 2k)θ Hn(x q) := e − , x = cos θ, (2.1) | X k k=0 q where n is the q-binomial coefficient, kq n (q; q)n = , (2.2) k q (q; q)k(q; q)n k − n 1 k and (a; q)0 = 1, (a; q)n = k=0− (1 aq ), n = 1, 2, 3, ... , is the q-shifted factorial. These polynomials can be generatedP by the− three-term recurrence relation n 2xHn(x q) = Hn+1(x q) + (1 q )Hn 1(x q), n 0, (2.3) | | − − | ≥ with the initial condition H0(x q) = 1. They have been found to enjoy many properties analogous to those known for the classical| Hermite polynomials [11, 12, 13, 14, 15]. In particular, the Rogers 1 generating function [11] for the q-Hermite polynomials has the form ∞ tn H (cos θ q) = e (teiθ)e (te iθ), t < 1, (2.4) X (q; q) n q q − n=0 n | | | where the q-exponential function eq(z) and its reciprocal Eq(z) are defined by n n(n 1)/2 ∞ z 1 ∞ q − n eq(z) := = (z; q)− ,Eq(z) := ( z) = (z; q) . (2.5) X (q; q) ∞ X (q; q) ∞ n=0 n n=0 n − It is often more convenient (cf.[4,5,6,7]) to make the change of variables x = cos θ xq(s) = sin κs π → in (2.4), which is equivalent to the substitution θ = 2 κs. That is to say, one can represent (2.4) as − ∞ tn g(s; t q) := H (sin κs q) = e (ite iκs)e ( iteiκs), t < 1. (2.4 ) X (q; q) n q − q 0 | n=0 n | − | | Observe that it is easy to verify (2.40) directly, by substituting the explicit form of n n k n i(2k n)κs Hn(sin κs q) := i ( 1) e − . (2.10) | X − k k=0 q into it and interchanging the order of summations with respect to the indices n and k. The advantage of such a parametrization x = cos θ = sin κs is that it actually incorporates both cases of the parameter q: 0 < q < 1 and q > 1. Indeed, to consider the case when q > 1 one | |1 | | | | may introduce the continuous q− -Hermite polynomials hn(x q) as [16] | n 1 hn(x q) := i− Hn(ix q− ). (2.6) | | The corresponding linear generating function for these polynomials is [17] ∞ qn(n 1)/2 − tnh (sinh κs q) = E (te κs)E ( teκs), (2.7) X (q; q) n q − q n=0 n | − where (cf.(2.10)) n k n k(k n) (n 2k)κs hn(sinh κs q) = ( 1) q − e − . (2.60) | X − k k=0 q From the inversion identity 1 1 n n(n+1)/2 (q− ; q− )n = ( 1) q− (q; q)n, n = 0, 1, 2,..., (2.8) − and the transformation property of the q-exponential functions (2.5)[18] e1/q(z) = Eq(qz), (2.9) it follows that ∞ qn(n+1)/2 g(s; t q 1) = ( it)nh (sinh κs q) = E (iqteκs)E ( iqte κs). (2.10) − X (q; q) n q q − | n=0 n − | − 2 1 1 The generating function g(s; iq− t q− ) thus coincides with the left-hand side of (2.7), so that (2.10) reproduces (2.7). | Examination of equations (2.40) and (2.10) reveals that the transformation of q 1/q actually → provides a reciprocal to the ((2.4)0) function 1 1 1 g− (s; t q) = g( is; q− t q− ). (2.11) | − | There is a second linear generating function n2/4 ∞ q n 2 f(s; t q) := t H (sin κs q) = E 2 (qt ) (sin κs; t) (2.12) X (q; q) n q q | n=0 n | E for the continuous q-Hermite polynomials [19, 6]. The q-exponential function q(x; t) in (2.12) is defined by [19] E n2/4 ∞ q 1 n 1 n 2 2 n − iκs − iκs (sin κs; t) := e 2 (qt )E 2 (t ) ( it) (q 2 e , q 2 e ; q) , (2.13) q q q X (q; q) − n E n=0 n − − k where (a1, . , ak; q)n = (aj; q)n is the conventional contracted notation for the multiple q- Qj=1 shifted factorials [20]. It is also expressible as a sum of two 2φ1 basic hypergeometric series, i.e., 2 2 2iκs 2iκs 2 2 q(sin κs; t) = eq2 (qt )Eq2 (t )2φ1(qe , qe− ; q; q , t ) E 1/4 2q t 2 2iκs 2 2iκs 3 2 2 + sin κs 2φ1(q e , q e− ; q ; q , t ). (2.14) 1 q − Introduced in [19] and further explored in [6, 7, 21], this q-analogue of the exponential function est on the q-quadratic lattice xq(s) = sin κs enjoys the property [19] 1/2 1/q(x; t) = q(x; q t). (2.15) E E − Therefore, as follows from (2.6), (2.8) and (2.12), 2 ∞ qn /4 f(s; t q 1) = ( iq1/2t)nh (sinh κs q) = e (qt2)f(is; q1/2t q). (2.16) − X (q; q) n q | n=0 n − | − | Equating coefficients of like powers of the parameter t on both sides of (2.16), we find the relation [n/2] n n (q− ; q)2k k(n k+1) hn(x q) = i− q − Hn 2k(ix q) (2.17) | X (q; q)k − | k=0 1 between the q-Hermite and q− -Hermite polynomials, in which [n/2] stands for the greatest integer not exceeding n/2. In view of the definition (2.6), an alternate form of this expression is [n/2] n 1 (q− ; q)2k k(n k+1) Hn(x q− ) = q − Hn 2k(x q). (2.18) | X (q; q)k − | k=0 3 Once the relations (2.17) and (2.18) are established, they can be checked by a direct calculation using the explicit sums (cf.(2.1)) 2n 2iθ 2iθ n 2 q− , e , e− 2 2 H2n(x q) = ( 1) (q; q )n3φ2 − − q ; q , (2.19a) | − q, 0 2n 2iθ 2iθ n 3 2 q− , qe , qe− 2 2 H2n+1(x q) = ( q)− (q ; q )n2x3φ2 − − q ; q , (2.19b) | − q3, 0 for the q-Hermite polynomials in terms of 3φ2 basic hypergeometric series [6].
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