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arXiv · 2512.11937

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

Liang-Jia Guo·Min-Jie Luo
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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.

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Liang-Jia Guo, Min-Jie Luo. 2025-12-12. Erd\'{e}lyi-type integrals for $F_K$ function and their $q$-analogues. https://arxiv.org/abs/2512.11937

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