Equations and series of hypergeometric type
General hypergeometric systems of differential equations associated with a sublattice \(B \in \mathbb{Z}^N\) are introduced by \textit{I. M. Gel'fand}, \textit{M. I. Graev} and \textit{A. V. Zelevinskij} [Dokl. Akad. Nauk SSSR 295, 14-19 (1987; Zbl 0661.22005)]. In the present paper, the author proposes a unified general theory of such hypergeometric systems and their \(q \)-analogues. For this purpose, one starts with \(N\) entire functions \(u_1, \dots, u_N\) on \(\mathbb{C}^N\) such that \(u_i (\alpha) u_j (\alpha - e_i) = u_j (\alpha) u_i (\alpha - e_j)\), where \(\{e_i\}\) is the standard basis in \(\mathbb{C}^N\). Then the linear operators \(\partial_i\) are defined by \(\partial_i x^\alpha = u_i (\alpha) x^{\alpha - e_i}\). Using these operators one can define general hypergeometric systems associated with \(B \in \mathbb{Z}^N\). The usual system is given in the case \(u_i (\alpha) = \alpha_i\), and \(q\)-analogue in the case \(u_i (\alpha) = [\alpha_i]_{q_i}\). The author writes down the reduced differential systems and their power series solutions.
scientific article; zbMATH DE number 846564