Commun. Korean Math. Soc. 2015; 30(3): 239-252
Printed July 31, 2015
https://doi.org/10.4134/CKMS.2015.30.3.239
Copyright © The Korean Mathematical Society.
Shyam Lal Kalla, Rakesh Kumar Parmar, and Sunil Dutt Purohit
Vyas Institute of Higher Education, Government College of Engineering and Technology, Rajasthan Technical University
Motivated mainly by certain interesting extensions of the $\tau$-hypergeometric function defined by Virchenko {\it et al.~}\cite{Vir-Ka-Za} and some $\tau$-Appell's function introduced by Al-Shammery and Kalla \cite{Al-Ka}, we introduce here the $\tau$-Lauricella functions $F_{A}^{ (n),\tau_{1},\ldots,\tau_{n}}$, $F_{B}^{ (n),\tau_{1},\ldots,\tau_{n}}$ and $F_{D}^{ (n),\tau_{1},\ldots,\tau_{n}}$ and the confluent forms $\Phi_{2}^{ (n),\tau_{1},\ldots,\tau_{n}}$ and $\Phi_{D}^{ (n),\tau_{1},\ldots,\tau_{n}}$ of $n$ variables. We then systematically investigate their various integral representations of each of these $\tau$-Lauricella functions including their generating functions. Various (known or new) special cases and consequences of the results presented here are also considered.
Keywords: generalized hypergeometric function, generlalized $\tau$-hyper\-geometric function, Appell's and Lauricella functions, $\tau$-Appell's function, $\tau$-Lauricella functions of several variables, generating function
MSC numbers: Primary 33C05, 33C15, 33C20; Secondary 33C65, 33C99
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