Communications of the
Korean Mathematical Society CKMS

Extensions of some classical special functions, for example, Beta function $B(x,y)$ and generalized hypergeometric functions $_{p}F_{q}$ have been actively investigated and found diverse applications. In recent years, several extensions for $B(x,y)$ and $_{p}F_{q}$ have been established by many authors in various ways. Here, we aim to generalize Appell's hypergeometric functions of two variables and Lauricella's hypergeometric function of three variables by using the extended generalized beta type function $B_{p}^{\left(\alpha ,\beta; m \right)} \left(x,y\right)$. Then some properties of the extended generalized Appell's hypergeometric functions and Lauricella's hypergeometric functions are investigated.

Keywords: beta function, extended generalized beta functions, extended generalized Gauss hypergeometric functions, extended generalized Appell's functions, extended generalized Lauricella's functions, Mellin transform

MSC numbers: Primary 33B15, 33B99, 33C05; Secondary 33C15, 33C20, 26A33