The Cauchy Operator for Basic Hypergeometric Series
Combinatorics
2007-08-21 v3
Abstract
We introduce the Cauchy augmentation operator for basic hypergeometric series. Heine's transformation formula and Sears' 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 . Using this operator, we obtain extensions of the Askey-Wilson integral, the Askey-Roy integral, Sears' two-term summation formula, as well as the -analogues of Barnes' lemmas. Finally, we find that the Cauchy operator is also suitable for the study of the bivariate Rogers-Szeg\"o polynomials, or the continuous big -Hermite polynomials.
Cite
@article{arxiv.0705.1812,
title = {The Cauchy Operator for Basic Hypergeometric Series},
author = {Vincent Y. B. Chen and Nancy S. S. Gu},
journal= {arXiv preprint arXiv:0705.1812},
year = {2007}
}