An Orthogonal Polynomial Solution to the Confluent-Type Heun’s Differential Equation

Saiful R. Mondal and Varun Kumar ()
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Saiful R. Mondal: Department of Mathematics and Statistics, College of Science, King Faisal University, P.O. Box 400, Al-Ahsa 31982, Saudi Arabia
Varun Kumar: Department of Pure and Applied Mathematics, Alliance School of Sciences, Alliance University, Bangalore 562106, Karnataka, India

Mathematics, 2025, vol. 13, issue 8, 1-18

Abstract: In this work, we present both analytical and numerical solutions to a seven-parameter confluent Heun-type differential equation. This second-order linear differential equation features three singularities: two regular singularities and one irregular singularity at infinity. First, employing the tridiagonal representation method (TRA), we derive an analytical solution expressed in terms of Jacobi polynomials. The expansion coefficients of the series are determined as solutions to a three-term recurrence relation, which is satisfied by a modified form of continuous Hahn orthogonal polynomials. Second, we develop a numerical scheme based on the basis functions used in the TRA procedure, enabling the numerical solution of the seven-parameter confluent Heun-type differential equation. Through numerical experiments, we demonstrate the robustness of this approach near singularities and establish its superiority over the finite difference method.

Keywords: continuous Hahn polynomials; confluent Heun’s differential equation; tridiagonal representation; recurrence relation; orthogonal polynomials; finite difference method; stability (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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