ABSTRACT

We consider several special functions defined by matrix integrals on Hermitian matrix space, namely, analogues of Gauss hypergeomeric, Kummer's confluent hypergeometric, Bessel, Hermite-Weber and Airy function. We discuss the relation of these functions of matrix integral type with some semi-classical orthogonal polynomials and with the polynomial solutions of quantum Painlevé equations.

中文翻译：

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矩阵超几何函数、半经典正交多项式和量子 Painlevé 方程

摘要

我们考虑由 Hermitian 矩阵空间上的矩阵积分定义的几个特殊函数，即 Gauss 超几何、Kummer 汇合超几何、Bessel、Hermite-Weber 和 Airy 函数的类似物。我们讨论了这些矩阵积分型函数与一些半经典正交多项式和量子Painlevé方程的多项式解的关系。