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

Liang-Jia Guo, Min-Jie Luo

TL;DR

The paper addresses Erdélyi-type integrals for Saran's multivariate $F_K$-function by delivering a new direct proof of a key Erdélyi-type integral and deriving a novel integral for Appell's $F_2$. It generalizes the framework to the $L$-variable $F_K$ (relevant in physics) and develops several $q$-analogs, including a Joshi–Vyas type theorem, a discrete $q$-analog, and a $q$-analogue of the main theorem, all supported by Dirichlet-type measures and basic hypergeometric functions. An Appendix catalogs known Erdélyi-type integrals across diverse hypergeometric functions, highlighting the historical and methodological landscape. Collectively, the results expand analytic tools for multivariate hypergeometric integrals and their $q$-deformations, with potential implications in mathematical physics, fractional calculus, and special function theory.

Abstract

In this paper, we revisit the recent result of Luo, Xu, and Raina [Fractal Fract. 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.

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

TL;DR

The paper addresses Erdélyi-type integrals for Saran's multivariate -function by delivering a new direct proof of a key Erdélyi-type integral and deriving a novel integral for Appell's . It generalizes the framework to the -variable (relevant in physics) and develops several -analogs, including a Joshi–Vyas type theorem, a discrete -analog, and a -analogue of the main theorem, all supported by Dirichlet-type measures and basic hypergeometric functions. An Appendix catalogs known Erdélyi-type integrals across diverse hypergeometric functions, highlighting the historical and methodological landscape. Collectively, the results expand analytic tools for multivariate hypergeometric integrals and their -deformations, with potential implications in mathematical physics, fractional calculus, and special function theory.

Abstract

In this paper, we revisit the recent result of Luo, Xu, and Raina [Fractal Fract. 6 (3) (2022)] on an Erdélyi-type integral for Saran's three-variable hypergeometric function . We provide a new proof of this integral and derive an attractive new integral related to Appell's function . A further extension on the -variable function, which appears in physics, is also discussed. Furthermore, we prove various -Erdélyi-type integrals for the -analogue of the -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.

Paper Structure

This paper contains 14 sections, 9 theorems, 99 equations.

Table of Contents

  1. Introduction
  2. Background and motivation
  3. The plan of the work
  4. Preliminaries
  5. Theorem \ref{['Th-ErdelyiIntegral-FK']} revisited
  6. An alternative proof of Theorem \ref{['Th-ErdelyiIntegral-FK']}
  7. A curious integral representation for $F_2$
  8. A further extension of Theorem \ref{['Th-ErdelyiIntegral-FK']}
  9. Some $q$-Erdélyi-type integrals
  10. A Joshi-Vyas type theorem
  11. A discrete $q$-analogue of Corollary \ref{['Cor-JoshVyas-generalization-2']}
  12. A $q$-analogue of Theorem \ref{['Th-ErdelyiIntegral-FK']}
  13. Conclusion
  14. Appendix

Key Result

Theorem 1.1

Let $\Re(\alpha_1+\eta_1)>\Re(\lambda_1)>0$, $\Re(\beta_2+\mu_2)>\Re(\lambda_2)>0$ and $\Re(\gamma_3)>\Re(\beta_1)>0$. Then we have where $(x,y,z) \in \mathbb{D}_K$ and $\mathrm{d}\mu_{\alpha,\beta}(t)$ is defined in DirichletMeasure.

Theorems & Definitions (21)

  • Theorem 1.1: Luo-Xu-Raina-2022
  • Remark 3.1
  • Theorem 3.2
  • proof
  • Remark 3.3
  • Theorem 3.4
  • proof
  • Theorem 4.1
  • proof
  • Remark 4.2
  • ...and 11 more