Series solutions of Heun-type equation in terms of orthogonal polynomials

Abstract

We introduce a nine-parameter Heun-type differential equation and obtain three classes of its solutions as series of square integrable functions written in terms of the Jacobi polynomial. The expansion coefficients of the series satisfy three-term recursion relations, which are solved in terms of orthogonal polynomials with continuous and/or discrete spectra. Some of these are well-known polynomials while others are either new or modified versions of known ones.

15 Pages, 3 Tables

References in corpus (6)

• Solution of the nonrelativistic wave equation in the tridiagonal representation approach
• Tridiagonalization and the Heun equation
• Open problem in orthogonal polynomials
• The Wilson-Racah Quantum System
• Orthogonal polynomials derived from the tridiagonal representation approach
• Series solutions of Laguerre- and Jacobi-type differential equations in terms of orthogonal polynomials and physical applications

Cited by in corpus (10)

• Open problem in orthogonal polynomials
• Painlevé IV, Chazy II, and Asymptotics for Recurrence Coefficients of Semi-classical Laguerre Polynomials and Their Hankel Determinants
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• Series solution of a ten-parameter second order differential equation with three regular and one irregular singularities
• Series solutions of Bessel-type differential equation in terms of orthogonal polynomials and physical applications
• Mapping Schrödinger equation into a Heun-type and identifying the corresponding potential function, energy and wavefunction
• Bound-states for generalized trigonometric and hyperbolic Pöschl-Teller potentials
• Solutions of the scattering problem in a complete set of Bessel functions with a discrete index
• Structural Algebraic Quantum Field Theory
• Progressive approximation of bound states by finite series of square-integrable functions