On higher Heine-Stieltjes polynomials

TL;DR

This paper investigates higher Heine-Stieltjes polynomials by analyzing a multiparameter spectral problem involving differential operators with polynomial coefficients, focusing on root distributions of solutions as parameters vary.

Contribution

It introduces a framework for studying polynomial solutions of differential equations with polynomial coefficients and computes root distributions for sequences of such solutions.

Findings

Root-counting measures for polynomial solutions are characterized.

The spectral problem extends classical Heine-Stieltjes theory to higher-order operators.

Convergence results for sequences of polynomial solutions are established.

Abstract

Given a differential operator T=\sum_{i=1}^k Q_i(z)d^i/dz^i where each Q_i(z) is a polynomial define r=max_i deg(Q_i(z)-i). Assuming that r is nonnegative we consider the following multiparameter spectral problem: for each positive integer n find all polynomials V(z) of degree at most r such that the equation T(S(z))+V(z)S(z)=0 has a polynomial solution S(z) of degree n. We calculate for any converging sequence of normalized polynomials V_j(z) the root-counting measure of the corresponding sequence of polynomials S_j(z).