Erd\'{e}lyi-type integrals for $F_K$ function and their $q$-analogues

TL;DR

This paper explores Erdélyi-type integrals for the hypergeometric function $F_K$ and its $q$-analogues, providing new proofs, extensions, and a comprehensive compilation of related integrals across various hypergeometric functions.

Contribution

It introduces new proofs and extensions of Erdélyi-type integrals for $F_K$ and its $q$-analogues, including a discrete analogue and a compilation of known integrals.

Findings

New proof of Erdélyi-type integral for $F_K$

Derived new integral related to Appell's $F_2$

Proved $q$-Erdélyi-type integrals for the $q$-analogue of $F_K$

Abstract

In this paper, we revisit the recent result of Luo, Xu, and Raina [Fractal Fract. 6 (3) (2022)] on an Erd\'{e}lyi-type integral for Saran's three-variable hypergeometric function $F_K$. We provide a new proof of this integral and derive an attractive new integral related to Appell's function $F_2$. A further extension on the $L$-variable $F_K$ function, which appears in physics, is also discussed. Furthermore, we prove various $q$-Erd\'{e}lyi-type integrals for the $q$-analogue of the $F_K$-function. An interesting discrete analogue is also included. We also provide a valuable compilation of the sources for known Erd\'{e}lyi-type integrals of many different hypergeometric functions in the Appendix.