The Cauchy operator for basic hypergeometric series

Abstract

We introduce the Cauchy augmentation operator for basic hypergeometric series. Heine's 2 φ 1 transformation formula and Sears' 3 φ 2 transformation formula can be easily obtained by the symmetric property of some parameters in operator identities. The Cauchy operator involves two parameters, and it can be considered as a generalization of the operator T (bD q ). Using this operator, we obtain extensions of the Askey-Wilson integral, the Askey-Roy integral, Sears' two-term summation formula, as well as the qanalogs of Barnes' lemmas. Finally, we find that the Cauchy operator is also suitable for the study of the bivariate Rogers-Szegö polynomials, or the continuous big q-Hermite polynomials.

A Note on q‐Integrals and Certain Generating Functions

“…If satisfying the difference equation then we have where and defined by , …”

q-Difference equation and the Cauchy operator identities

“…We recall that Chen and Gu [6] introduced the Cauchy operator (1.6) as the basis of parameter augmentation which serves as a method for proving extensions of the Askey-Wilson integral, the Askey-Roy integral and so on. Liu [12] established two general q-exponential operator identities by solving two simple q-difference equations.…”

“…In an attempt to find efficient q-shift operators to deal with basic hypergeometric series identities in the framework of the q-umbral calculus [1,2,10], Chen and Liu [7,8] introduced two q-exponential operators, Fang [9] introduced a new q-exponential operator, Chen and Gu [6] introduced a Cauchy operator for deriving identities from their special cases. In this paper, motivated by their work, we study some applications of the Cauchy operator for basic hypergeometric series.…”

An operator approach to the Al-Salam–Carlitz polynomials

“…This paper presents an operator approach to the Rogers-type formulas and Mehler's formulas for the Al-Salam-Carlitz polynomials. These polynomials are a generalization of the classical Rogers-Szegö polynomials which have been extensively studied, see for example [6][7][8]10,13]. There are two classical formulas concerning the RogersSzegö polynomials, namely, Mehler's formula and the Rogers formula, in connection with the Poisson kernel formula and the linearization formula.…”

“…On the other hand, although the Al-Salam-Carlitz polynomials can be expressed in terms of the bivariate Rogers-Szegö polynomials 10) as noted by Carlitz [8], it is often happens that an infinite q-series identity no longer holds when q is replaced by q −1 . In fact, it turns out to be the case for the Rogers formula and Mehler's formula for the polynomials h n (x, y|q).…”