On zeros of polynomials associated with Heun class equations

TL;DR

This work develops a perturbative framework for the accessory-parameter problem in Heun-class equations by studying the zeros of the polynomials $c_{m+1}(B)$ that arise in local solution expansions. Using a three-term recurrence and small-$s$ expansions, the authors derive explicit asymptotics for zeros and prove that zeros of the limiting coefficient $d_2(B)$ are approached by zeros of $c_{m+1}(B)$ as $m\to\infty$, linking local data to global spectral structure. The framework is applied to Lamé, Mathieu, and Whittaker–Hill equations, with substantial numerical evidence that these zeros reproduce key spectral features and reveal parts of the spectrum for these equations. Together, the results provide a bridge between local series coefficients and global spectral information for Heun-class systems and offer a practical approach to approximate eigen-spectra in settings inspired by AGT and finite-gap theory.

Abstract

Schäfke and Schmidt established that the asymptotics of the coefficients of the local solution to some linear differential equation is related to global structures of solutions. The Heun class equations have the accessory parameters, and we investigate the polynomials whose variable is the accessory parameter which appears as the coefficients of the local solution. By calculating the zeros of the polynomials numerically, we obtain the data of the spectral related to the Heun class equations numerically.

Theorems & Definitions (15)

• Theorem 2.1
• Proposition 4.1
• Proposition 4.2
• Proposition 4.3
• proof
• Proposition 4.4
• proof
• Proposition 4.5
• proof
• Theorem 4.6