2011/12/21 by Zhuangchu Luo, Hua Chen, Luo, Zhuangchu +3
Mathematics · #33C65 #Advanced Differential Equations and Dynamical Systems #Advanced Mathematical Identities #Analysis of PDEs (math.AP) #FOS: Mathematics #Mathematical functions and polynomials #math.AP #msc:33C65
paper · pdf · doi:10.48550/arxiv.1112.4981
arxiv created 2011/12/21 · openalex publication_date 2011/12/21 · arxiv updated 2011/12/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/04
This paper is devoted to the family \Gn\ of hypergeometric series of any finite number of variables, the coefficients being the square of the multinomial coefficients (ℓ1+...+ℓn)!/(ℓ1!...ℓn!), where n∈\ZZ≥ 1. All these series belong to the family of the general Appell-Lauricella's series. It is shown that each function Gn can be expressed by an integral involving the previous one, Gn-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 Gn with regard to the action of a finite number of involutions of \CC*n is also established. Finally, a particular attention is reserved to the study of the functions G2 and G3, each of which is proved to be algebraic or to be expressed by the Legendre's elliptic function of the first kind.