Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A formula for the local Heun solution

Pavel Šťovíček

TL;DR

This work derives explicit, finite-sum representations for the coefficients of the local Heun solution, the unique Frobenius-type solution analytic in the unit disk and normalized by $F(0)=1$. By marrying the theory of orthogonal polynomials with Green functions of Jacobi matrices, the authors express the coefficients $c_n$ through structured sums tied to orthonormal polynomial sequences and associated three-term recurrences, enabling both exact formulas and quantitative bounds. A diagonal perturbation framework extends the construction to general $\delta$, and two concrete realizations are presented: a general five-parameter subfamily with a detailed coefficient formula, and a special subfamily with $\delta=\beta+1$ that furnishes an alternate, compatible representation and a parallel bound. The results provide insight into the asymptotic behavior of the coefficients and offer practical, explicit expressions for computing the local Heun solution in terms of orthogonal-polynomial data.

Abstract

The local Heun solution is the unique solution to Heun's equation which is analytic in the unit disk centered at $0\in\mathbb{C}$ and taking the value $1$ at the center of the disk. In this paper, as an application of the theory of orthogonal polynomials, we are able to express the coefficients in the corresponding power series as finite multiple sums. In addition, the obtained formula can be used to derive an explicit estimate on the coefficients giving a hint on their asymptotic behavior for large indices.

A formula for the local Heun solution

TL;DR

This work derives explicit, finite-sum representations for the coefficients of the local Heun solution, the unique Frobenius-type solution analytic in the unit disk and normalized by . By marrying the theory of orthogonal polynomials with Green functions of Jacobi matrices, the authors express the coefficients through structured sums tied to orthonormal polynomial sequences and associated three-term recurrences, enabling both exact formulas and quantitative bounds. A diagonal perturbation framework extends the construction to general , and two concrete realizations are presented: a general five-parameter subfamily with a detailed coefficient formula, and a special subfamily with that furnishes an alternate, compatible representation and a parallel bound. The results provide insight into the asymptotic behavior of the coefficients and offer practical, explicit expressions for computing the local Heun solution in terms of orthogonal-polynomial data.

Abstract

The local Heun solution is the unique solution to Heun's equation which is analytic in the unit disk centered at and taking the value at the center of the disk. In this paper, as an application of the theory of orthogonal polynomials, we are able to express the coefficients in the corresponding power series as finite multiple sums. In addition, the obtained formula can be used to derive an explicit estimate on the coefficients giving a hint on their asymptotic behavior for large indices.
Paper Structure (8 sections, 6 theorems, 87 equations)

This paper contains 8 sections, 6 theorems, 87 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Green function
  4. Generalities
  5. The particular case $\delta=0$
  6. A diagonal perturbation of a Jacobi matrix
  7. The local Heun solution
  8. The five-parameter subfamily for $\delta=\beta+1$

Key Result

Proposition 1

The Green function for the Jacobi parameters $a_{n}$, $b_{n}$ given in (eq:alpha-beta-delta-0) reads and $G_{m,n}=0$ for $0\le m\le n$.

Theorems & Definitions (13)

  • Proposition 1
  • Theorem 2
  • Remark
  • proof
  • Theorem 3
  • Remark
  • proof
  • Proposition 4
  • proof
  • Theorem 5
  • ...and 3 more