On a family of symmetric hypergeometric functions of several variables and their Euler type integral representation

This paper is devoted to the family {_Gn}$\{G_n\}$ of hypergeometric series of any finite number of variables, the coefficients being the square of the multinomial coefficients (_?1+?+_?n)!/(_?1!…_?n!)$({?}_1+?+{?}_n)!/({?}_1!\dots {?}_n!)$, where n∈_Z?1$n\in {\mathbb{Z}}_{?1}$. All these series belong to the family of the general Appell–Lauricella?s series. It is shown that each function _Gn$G_n$ can be expressed by an integral involving the previous one, _Gn−1$G_{n-1}$. Thus this family can be represented by a multidimensional Euler type integral, what suggests some explicit link with the Gelfand–Kapranov–Zelevinsky?s theory of A  -hypergeometric systems or with the Aomoto?s theory of hypergeometric functions. The quasi-invariance of each function _Gn$G_n$ with regard to the action of a finite number of involutions of C^?n${\mathbb{C}}^{?n}$ is also established. Finally, a particular attention is reserved to the study of the functions _G2$G_2$ and _G3$G_3$, each of which is proved to be algebraic or to be expressed by the Legendre?s elliptic function of the first kind.