Analytic continuation of Lauricella's function F_D^(N) for large in modulo variables near hyperplanes {z_j = z_l}

Abstract

We consider the Lauricella hypergeometric function $F_D^{(N)}$, depending on $N\ge 2$ variables $z_1,\dots ,z_N$, and obtain formulas for its analytic continuation into the vicinity of a singular set which is an intersection of the hyperplanes $\{z_j=z_l\}$. It is assumed that all N variables are large in modulo. This formulas represent the function $F_D^{(N)}$ outside of the unit polydisk in the form of linear combinations of other N-multiple hypergeometric series that are solutions of the same system of partial differential equations as $F_D^{(N)}$. The derived hypergeometric series are N-dimensional analogues of the Kummer solutions that are well known in the theory of the classical hypergeometric Gauss equation.

Keywords:

• Multiple hypergeometric functions
• Lauricella functions
• analytic continuation
• Horn functions
• PDEs system of equations

AMS CLASSIFICATIONS:

• 33C65
• 32D15
• 33C90

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work is supported by the Ministry of Science and Higher Education of the Russian Federation: agreement no. 075-03-2020-223/3 (FSSF-2020-0018).