Junesang Choi, Anvar Hasanov, and Mamasali Turaev Dongguk University, Dongguk University, Dongguk University

Abstract : Exton introduced 20 distinct triple hypergeometric functions whose names are $X_i$ $(i=1,\,\ldots,\,20)$ to investigate their twenty Laplace integral representations whose kernels include the confluent hypergeometric functions ${}_0F_1$, ${}_1F_1$, a Humbert function $\Psi_1$, and a Humbert function $\Phi_2$. The object of this paper is to present 18 new integral representations of Euler type for the Exton hypergeometric function $X_8$, whose kernels include the Exton functions ($X_2$, $X_8$) itself, the Horn's function $H_4$, the Gauss hypergeometric function $F$, and Lauricella hypergeometric function $F_C$. We also provide a system of partial differential equations satisfied by $X_8$.

Keywords : generalized hypergeometric series, multiple hypergeometric functions, integrals of Euler type, Laplace integral, Exton functions $X_i$, Humbert functions, Appell-Horn function $H_4$, Lauricella hypergeometric function $F_C$