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EXTENSIONS OF MULTIPLE LAURICELLA AND HUMBERT’S CONFLUENT HYPERGEOMETRIC FUNCTIONS THROUGH A HIGHLY GENERALIZED POCHHAMMER SYMBOL AND THEIR RELATED PROPERTIES

최준상|Naveen Kumar|Rakesh K. Parmar|RITU AGARWAL
대한수학회 (The Korean Mathematical Society)|2023.05|대한수학회보 (Bulletin of the KMS)|제60권 제3호|pp. 575-591|DOI 10.4134/BKMS.b210652

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키워드

Laguerre polynomialsAppell functionsincomplete gamma functionsLauricella functionsGeneralized gamma functionsgeneralized Pochhammer symbolgeneralized Gauss hypergeometric functionBessel and modified Bessel functionsgeneralized multivariable Lauricella functions
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초록

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.

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.

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Extended Lauricella and Appell Functions and Their Associated PropertiesH. M. Srivastava, M. M. Joshi, R. K. Parmar장전수학회 · Advanced Studies in Contemporary Mathematics2015 · 제25권 제2호The Incomplete Lauricella Functions of Several Variables and Associated Properties and Formulas최준상, H. M. Srivastava, Rakesh K. Parmar경북대학교 자연과학대학 수학과 · Kyungpook Mathematical Journal2018 · 제58권 제1호Some $\tau$-extensions of Lauricella functions of several variablesRakesh Kumar Parmar, Shyam Lal Kalla, Sunil Dutt Purohit대한수학회 · 대한수학회논문집2015 · 제30권 제3호Generalized Incomplete Pochhammer Symbols and Their Applications to Hypergeometric functionsAshish Verma, Vivek Sahai경북대학교 자연과학대학 수학과 · Kyungpook Mathematical Journal2018 · 제58권 제1호Extended beta, hypergeometric and confluent hypergeometric functions via multi-index mittag-leffler functionMohd Aman, Nabiullah Khan, Talha Usman장전수학회 · Proceedings of the Jangjeon Mathematical Society(장전수학회 논문집)2022 · 제25권 제1호Extended hypergeometric functions of two and three variables최준상, Praveen Agarwal, Shilpi Jain대한수학회 · 대한수학회논문집2015 · 제30권 제4호

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