Some $\tau$-extensions of Lauricella functions of several variables
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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.
Abstract
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.
참고문헌 (13)
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