New integral results improve understanding of hypergeometric functions and their q-analogues, indicating potential applications in physics.
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 FK. We provide a new proof of this integral and derive an attractive new integral related to Appell's function F₂. A further extension on the L-variable FK function, which appears in physics, is also discussed. Furthermore, we prove various q-Erdélyi-type integrals for the q-analogue of the FK-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.
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Guo et al. (2025) studied this question.
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