Closed-form solution of a general three-term recurrence relation: applications to Heun functions and social choice models

TL;DR

This paper presents a new closed-form solution for three-term recurrence relations, applicable to diverse fields like quantum mechanics, biology, economics, and social choice, simplifying calculations and analysis.

Contribution

It introduces a linear algebraic method to solve three-term recurrence relations without continued fractions, enabling explicit solutions in terms of orthogonal polynomials.

Findings

Provides closed-form solutions for Heun function coefficients

Derives equations for relaxation times in social choice models

Simplifies analysis of recurrence relations in various scientific contexts

Abstract

We derive a concise closed-form solution for a linear three-term recurrence relation. Such recurrence relations are very common in the quantitative sciences, and describe finite difference schemes, solutions to problems in Markov processes and quantum mechanics, and coefficients in the series expansion of Heun functions and other higher-order functions. Our solution avoids the usage of continued fractions and relies on a linear algebraic approach that makes use of the properties of lower-triangular and tridiagonal matrices, allowing one to express the terms in the recurrence relation in closed-form in terms of a finite set of orthogonal polynomials. We pay particular focus to the power series coefficients of Heun functions, which are often found as solutions in eigenfunction problems in quantum mechanics and general relativity and have also been found to describe time-dependent dynamics in both biology and economics. Finally, we apply our results to find equations describing the relaxation times to steady state behaviour in social choice models.