Hermite-Weber and Airy Functions of Matrix Argument and their Holonomic Systems

Keimur H and Lia C

Published on: 2023-01-30

Abstract

We consider several special functions defined by matrix integrals on Hermitian matrix space. They are analogues of Gauss hypergeomeric, Kummer’s confluent hypergeometric, Bessel, Hermite-Weber and Airy functions. We give the systems of differential equations for these analogues of classical special functions and discuss their holonomicity and give their holonomic ranks.

Keywords

Hypergeometric function; Matrix integral; Holonomic system; Grobner basis 2020 Mathematics Subject Classification Numbers; 33C70; 33C80

Introduction

We are interested in the hypergeometric functions (HGF) of a matrix argument defined by integrals on the space of complex Hermitian matrices. One of the most important class of classical special functions is the Gauss HGF and its confluent family, namely, Kummer’s confluent HGF, Bessel function, Hermite-Weber function and Airy function.