Related papers: An expansion formula for the Askey-Wilson function
We are interested in the asymptotic behavior of orthogonal polynomials of the generalized Jacobi type as their degree $n$ goes to $\infty$. These are defined on the interval $[-1,1]$ with weight function…
A q-analogue of Erdelyi's formula for the Hankel transform of the product of Laguerre polynomials is derived using the quantum linking groupoid between the quantum SU(2) and E(2) groups. The kernel of the q-Hankel transform is given by the…
The little and big q-Jacobi polynomials are shown to arise as basis vectors for representations of the Askey-Wilson algebra. The operators that these polynomials respectively diagonalize are identified within the Askey-Wilson algebra…
We classify the shift operators for the symmetric Askey-Wilson polynomials and construct shift operators for the non-symmetric Askey-Wilson polynomials using two decompositions of non-symmetric Askey-Wilson polynomials in terms of symmetric…
The aim of this work is to study new functions arising from the limit transition of the Jackson's $q$-Bessel functions when $q\rightarrow -1$. These functions coincide with the $cas$ function for particular values of their parameters. We…
Considering the kernel of an integral operator intertwining two realizations of the group of motions of the pseudo-Euclidian space, we derive two formulas for series containing Whittaker's functions or Weber's parabolic cylinder functions.…
In 1992 A. Zhedanov introduced the Askey-Wilson algebra AW=AW(3) and used it to describe the Askey-Wilson polynomials. In this paper we introduce a central extension $\Delta$ of AW, obtained from AW by reinterpreting certain parameters as…
In this paper we return to the study of the Watson kernel for the Abel summabilty of Jacobi polynomial series. These estimates have been studied for over more than 30 years. The main innovations are in the techniques used to get the…
The modified q-Bessel functions and the q-Bessel-Macdonald functions of the first and second kind are introduced. Their definition is based on representations as power series. Recurrence relations, the q-Wronskians, asymptotic…
A construction is given of the most general representations of the q-oscillator algebra where both generators are tridiagonal. It is shown to be connected to the Askey-Wilson polynomials.
Simultaneous eigenfunctions of two Askey-Wilson second order difference operators are constructed as formal matrix coefficients of the principal series representation of the modular double of the quantized universal enveloping algebra…
A one-parameter family of operators that have the complementary Bannai-Ito (CBI) polynomials as eigenfunctions is obtained. The CBI polynomials are the kernel partners of the Bannai-Ito polynomials and also correspond to a $q\rightarrow-1$…
We construct a commutative algebra A_z, generated by d algebraically independent q-difference operators acting on variables z_1, z_2,..., z_d, which is diagonalized by the multivariable Askey-Wilson polynomials P_n(z) considered by Gasper…
An analogue of Taylor's formula, which arises by substituting the classical derivative by a divided difference operator of Askey-Wilson type, is developed here. We study the convergence of the associated Taylor series. Our results…
We construct a non-polynomial generalization of the $q$-Askey scheme. Whereas the elements of the $q$-Askey scheme are given by $q$-hypergeometric series, the elements of the non-polynomial scheme are given by contour integrals, whose…
We use orthogonality measures of Askey--Wilson polynomials to construct Markov processes with linear regressions and quadratic conditional variances. Askey--Wilson polynomials are orthogonal martingale polynomials for these processes.
The elliptic gamma function is a generalization of the Euler gamma function. Its trigonometric and rational degenerations are the Jackson q-gamma function and the Euler gamma function. We prove multiplication formulas for the elliptic gamma…
In the lecture notes we start off with an introduction to the $q$-hypergeometric series, or basic hypergeometric series, and we derive some elementary summation and transformation results. Then the $q$-hypergeometric difference equation is…
The sieved Jacobi polynomials have been introduced by Askey. These can be obtained from conveniently taking $q$ to be a root of unity in the Askey-Wilson polynomials. The question of determining if they are eigenfunctions of some operator…
We present a one-parameter family of constant solutions of the reflection equation and define a family of quantum complex Grassmannians endowed with a transitive action of the quantum unitary group. By computing the radial part of a…