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

This paper is devoted to the family {Gn}{Gn} of hypergeometric series of any finite number of variables, the coefficients being the square of the multinomial coefficients (ℓ1+⋯+ℓn)!/(ℓ1!…ℓn!)(ℓ1+⋯+ℓn)!/(ℓ1!…ℓn!), where n∈Z⩾1n∈Z⩾1. All these series belong to the family of the general Appell–Lauricellaʼs series. It is shown that each function GnGn can be expressed by an integral involving the previous one, Gn−1Gn−1. Thus this family can be represented by a multidimensional Euler type integral, what suggests some explicit link with the Gelfand–Kapranov–Zelevinskyʼs theory of A  -hypergeometric systems or with the Aomotoʼs theory of hypergeometric functions. The quasi-invariance of each function GnGn with regard to the action of a finite number of involutions of C⁎nC⁎n is also established. Finally, a particular attention is reserved to the study of the functions G2G2 and G3G3, each of which is proved to be algebraic or to be expressed by the Legendreʼs elliptic function of the first kind.