Communications of the
Korean Mathematical Society CKMS

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