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This paper is devoted to the family $\{G_n\}$ of hypergeometric series of anyfinite number of variables, the coefficients being the square of themultinomial coefficients $(\ell_1+...+\ell_n)!/(\ell_1!...\ell_n!)$, where$n\in\ZZ_{\ge 1}$. All these series belong to the family of the generalAppell-Lauricella's series. It is shown that each function $G_n$ can beexpressed by an integral involving the previous one, $G_{n-1}$. Thus thisfamily can be represented by a multidimensional Euler type integral, whatsuggests some explicit link with the Gelfand-Kapranov-Zelevinsky's theory of$A$-hypergeometric systems or with the Aomoto's theory of hypermeotricfunctions. The quasi-invariance of each function $G_n$ with regard to theaction of a finite number of involutions of $\CC^{*n}$ 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 theLegendre's elliptic function of the first kind.

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