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

This paper is devoted to the family $\{G_n\}$ of hypergeometric series of any finite number of variables, the coefficients being the square of the multinomial coefficients $({\mathrm{\ell }}_1+\mathrm{.}\mathrm{.}\mathrm{.}+{\mathrm{\ell }}_n)!\mathrm{/}({\mathrm{\ell }}_1!\mathrm{.}\mathrm{.}\mathrm{.}{\mathrm{\ell }}_n!)$, where $n\in {\text{ZZ}}_{\ge 1}$. All these series belong to the family of the general Appell-Lauricella’s series. It is shown that each function $G_n$ can be expressed by an integral involving the previous one, $G_{n-1}$. 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 hypermeotric functions. The quasi-invariance of each function $G_n$ with regard to the action of a finite number of involutions of ${\text{CC}}^{\ast n}$ is also established. Finally, a particular attention is reserved to the study of the functions $G_2$ and $G_3$, each of which is proved to be algebraic or to be expressed by the Legendre’s elliptic function of the first kind.