Erdélyi-type integrals for $F_K$ function and their $q$-analogues

Abstract: In this paper, we revisit the recent result of Luo, Xu, and Raina [Fractal Frac. 6 (3) (2022)] on an Erdélyi-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 $q$-Erdélyi-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 Erdélyi-type integrals of many different hypergeometric functions in the Appendix.