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

TL;DR

This work derives a closed-form solution for a general three-term recurrence by transforming the problem into an infinite-dimensional linear system with a lower-triangular matrix and solving via matrix inversion. The key advance is expressing $C_i$ in terms of recursively defined orthogonal polynomials: $C_i = \dfrac{\tilde{Q}_0 \phi^{i-1}_{i-1}(0) - R_0 P_1 \phi^{i-1}_{i-2}(0)}{\prod_{j=0}^{i-1}R_j}$, with connections to continued fractions and a clear parallel to Risken’s methods. The framework applies to Heun function Frobenius coefficients, enabling closed-form series coefficients, and extends to practical relaxation-time calculations in social-choice models, yielding polynomial conditions for eigenvalues. Overall, the approach provides a unified, analytic pathway to obtain spectral data and Frobenius coefficients for a broad class of problems in physics, biology, economics, and beyond. The results advance analytic tractability in master equations and higher-order special functions, with potential extensions to higher-order recurrences and reaction networks.

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.

Figures (1)

• Figure 1: Illustration showing the radii of convergence of the Frobenius solutions of the general Heun equation about $z=0,1$ and $a$. The parameter $a$ can take any complex value. Each Frobenius solution is valid in a circle in the complex plane (centred at the respective singularity) valid until the next singularity. Ordinary series expansions, not about singularities, are also valid in a circle extending up to the next singularity. Note that in cases where the Heun functions simplify to polynomials, or even Frobenius solutions valid at two singularities, the radii of convergence will extend beyond those in the illustration. The radius of convergence of the solution at $z=\infty$ can be seen clearly through the independent variable transformation $z\to1/x$, as shown in ronveaux1995heun. For further details see motygin2015numerical.