Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

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Bull. Korean Math. Soc. 2023; 60(3): 575-591

Online first article May 11, 2023      Printed May 31, 2023

https://doi.org/10.4134/BKMS.b210652

Copyright © The Korean Mathematical Society.

Extensions of multiple Lauricella and Humbert's confluent hypergeometric functions through a highly generalized Pochhammer symbol and their related properties

Ritu Agarwal, Junesang Choi, Naveen Kumar, Rakesh K. Parmar

Malaviya National Institute of Technology; Dongguk University; Malaviya National Institute of Technology; Pondicherry University-A Central University

Abstract

Motivated by several generalizations of the Pochhammer \linebreak symbol and their associated families of hypergeometric functions and hypergeometric polynomials, by choosing to use a very generalized Pochhammer symbol, we aim to introduce certain extensions of the generalized Lauricella function $F_A^{(n)}$ and the Humbert's confluent hypergeometric function $\Psi^{(n)}$of $n$ variables with, as their respective particular cases, the second Appell hypergeometric function $F_2$ and the generalized Humbert's confluent hypergeometric functions $\Psi_2$ and investigate their several properties including, for example, various integral representations, finite summation formulas with an $s$-fold sum and integral representations involving the Laguerre polynomials, the incomplete gamma functions, and the Bessel and modified Bessel functions. Also, pertinent links between the major identities discussed in this article and different (existing or novel) findings are revealed.

Keywords: Generalized gamma functions, generalized Pochhammer symbol, generalized Gauss hypergeometric function, Lauricella functions, Appell functions, Laguerre polynomials, incomplete gamma functions, Bessel and modified Bessel functions, generalized second Appell function in two variables, generalized multivariable Lauricella functions, generalized multivariable Humbert's confluent hypergeometric function

MSC numbers: Primary 33B15, 33B20, 33C05, 33C15, 33C20; Secondary 33B99, 33C99, 60B99

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