A Ramanujan-type measure for the Askey-Wilson polynomials

A Ramanujan-type representation for the Askey-Wilson q-beta integral, admitting the transformation q to q(exp -1), is obtained. Orthogonality of the Askey-Wilson polynomials with respect to a measure, entering into this representation, is proved. A simple way of evaluating the Askey-Wilson q-beta integral is also given

Kravchuk functions for the finite oscillator approximation

Kravchuk orthogonal functions - Kravchuk polynomials multiplied by the square root of the weight function - simplify the inversion algorithm for the analysis of discrete, finite signals in harmonic oscillator components. They can be regarded as the best approximation set. As the number of sampling points increases, the Kravchuk expansion becomes the standard oscillator expansion

Una Función Generatriz no Estandar para Polinomios q-Hahn Duales Continuos

We study a non-standard form of generating function for the three-parameter continuous dual q-Hahn polynomials $p_{n} (x; a, b, | q)$, which has surfaced in a recent work of the present authors on the construction of lifting $q$-difference operators in the Askey scheme of basic hypergeometric polynomials. We show that the resulting generating function identity for the continuous dual q-Hahn polynomials $p_{n} (x; a, b, c | q)$ can be explicitly stated in terms of Jackson’s $q$-exponential functions $e_{q} (z)$.Estudiamos una forma no estándar de la función generatriz para una familia de polinomios duales continuos -Hahn de tres parámetros , que han surgido en un trabajo reciente de los autores en la construcción de operadores elevadores en -diferencias del esquema de Askey de polinomios básicos hipergeométricos. Demostramos que la función generatriz identidad resultante para los polinomios q-Hahn duales continuos puede ser expresada explícitamente en términos de las funciones -exponenciales de Jackson

On a q-extension of the Hermite polynomials Hn(x) with the continuous orthogonality property on R1

We study a polynomial sequence of q-extensions of the classical Hermite polynomials Hn(x), which satisfi es continuous orthogonality on the whole real line R with respect to the positive weight function. This sequence can be expressed either in terms of the q-Laguerre polynomials L n (x; q), = 1=2, or through the discrete q-Hermite polynomials ~hn(x; q) of type II.Ministerio de Ciencia y TecnologíaJunta de AndalucíaUnión EuropeaDirección General Asuntos del Personal Académico (Universidad Nacional Autónoma de México

On a q-extension of the linear harmonic oscillator with the continuous orthogonality property on R

We discuss a q-analogue of the linear harmonic oscillator in quantum mechanics, based on a q-extension of the classical Hermite polynomials Hn(x), recently introduced by us in [1] R. Alvarez-Nodarse, M. K. Atakishiyeva, and N. M. Atakishiyev.. On a q-extension of the Hermite polynomials Hn(x) with the continuous orthogonality property on R. Boletín de la Sociedad Matemática Mexicana (3), 8, No.2, pp.127–139, 2002. The wave functions in this q-model of the quantum harmonic oscillator possess the continuous orthogonality property on the whole real line R with respect to a positive weight function. A detailed description of the corresponding q-system is carried out.Dirección General de Enseñanza SuperiorJunta de AndalucíaDirección General Asuntos del Personal Académico (Universidad Nacional Autónoma de México

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

In this paper we study in detail a q-extension of the generalized Hermite polynomials of Szeg˝o. A continuous orthogonality property on R with 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.Ministerio de Ciencia y TecnologíaJunta de AndalucíaDirección General Asuntos del Personal Académico (Universidad Nacional Autónoma de México

Mellin transforms for some families of q-polynomials

By using Ramanujan's q-extension of the Euler integral representation for the gamma function, we derive the Mellin integral transforms for the families of the discrete q-Hermite II, the Al-Salam–Carlitz II, the big q-Laguerre, the big q-Legendre, the big q-Jacobi and the q-Hahn polynomials.Ministerio de Ciencia y TecnologíaJunta de AndalucíaDirección General de Asuntos del Personal Académico (Universidad Nacional Autónoma de México)UNAM IN11230