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We derive a concise closed-form solution for a linear three-term recurrencerelation. Such recurrence relations are very common in the quantitativesciences, and describe finite difference schemes, solutions to problems inMarkov processes and quantum mechanics, and coefficients in the seriesexpansion of Heun functions and other higher-order functions. Our solutionavoids the usage of continued fractions and relies on a linear algebraicapproach that makes use of the properties of lower-triangular and tridiagonalmatrices, allowing one to express the terms in the recurrence relation inclosed-form in terms of a finite set of orthogonal polynomials. We payparticular focus to the power series coefficients of Heun functions, which areoften found as solutions in eigenfunction problems in quantum mechanics andgeneral relativity and have also been found to describe time-dependent dynamicsin both biology and economics. Finally, we apply our results to find equationsdescribing the relaxation times to steady state behaviour in social choicemodels.

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