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

TL;DR

This paper introduces a nine-parameter Heun-type differential equation and derives series solutions using orthogonal polynomials, revealing new and modified polynomial solutions with specific recursion relations.

Contribution

It presents a novel nine-parameter Heun-type equation and constructs series solutions in terms of orthogonal polynomials, including new and modified polynomial families.

Findings

Series solutions expressed with Jacobi and other orthogonal polynomials

Recursion relations solved via orthogonal polynomials with continuous/discrete spectra

Identification of new or modified orthogonal polynomial solutions

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.