Isomonodromy sets of accessory parameters for Heun class equations

TL;DR

This paper explores the isomonodromy sets of accessory parameters for Heun class equations, linking them to Painlevé equations and deriving asymptotic approximations for these parameters.

Contribution

It establishes a connection between Heun class equations and Painlevé equations through limits, describing accessory parameters via Painlevé functions and tau functions.

Findings

Accessory parameters characterized by Painlevé function coefficients

Asymptotic approximations derived for various Heun equations

Link established between Heun equations and Painlevé linear systems

Abstract

In this paper, we consider the monodromy and, in particularly, the isomonodromy sets of accessory parameters for the Heun class equations. We show that the Heun class equations can be obtained as limits of the linear systems associated with the Painlev\'{e} equations when the Painlev\'e transcendents go to one of the actual singular points of the linear systems. While the isomonodromy sets of accessory parameters for the Heun class equations are described by the Taylor or Laurent coefficients of the corresponding Painlev\'{e} functions, or the associated tau functions, at the positions of the critical values. As an application of these results, we derive some asymptotic approximations for the isomonodromy sets of accessory parameters in the Heun class equations, including the confluent Heun equation, the doubly-confluent Heun equation and the reduced biconfluent Heun equation.