Analytic continuation of Lauricella's function F_D^(N) for variables close to unit near hyperplanes {z_j = z_l}

Abstract

For the Lauricella hypergeometric function $F_D^{(N)}$ with an arbitrary number of variables $z_1,\dots ,z_N$, we construct formulas for analytic continuation into the vicinity of hyperplanes $\{z_j=z_l\}$ and their intersections providing that all variables are close to unit. Such formulas represent the function $F_D^{(N)}$ near the point $(1,\dots ,1)$ as linear combinations of N–multiple hypergeometric series that are solutions of the same system of partial differential equations as $F_D^{(N)}$. Such series are the N–dimensional analog of the Kummer solutions known for the Gauss equation. The constructed analytical continuation formulas allow one to effectively calculate the function $F_D^{(N)}$ outside the unit polydisk.

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 Russian Foundation for Basic Research (Proj. 19-07-00750A).