Series solutions of confluent Heun equations in terms of incomplete Gamma-functions

TL;DR

This paper introduces a systematic method to express solutions of confluent Heun equations using incomplete Gamma-functions, providing new series expansions with specific recurrence relations and conditions for finite-sum solutions.

Contribution

A novel algorithm for constructing series solutions of confluent Heun equations in terms of incomplete Gamma-functions, including analysis of recurrence relations and finite-sum conditions.

Findings

Series expansions involve four- or five-term recurrence relations.

Conditions identified for finite-sum (closed-form) solutions.

Method applicable to various confluent Heun equations.

Abstract

We present a simple systematic algorithm for construction of expansions of the solutions of ordinary differential equations with rational coefficients in terms of mathematical functions having indefinite integral representation. The approach employs an auxiliary equation involving only the derivatives of a solution of the equation under consideration. Using power-series expansions of the solutions of this auxiliary equation, we construct several expansions of the four confluent Heun equations' solutions in terms of the incomplete Gamma-functions. In the cases of single- and double-confluent Heun equations the coefficients of the expansions obey four-term recurrence relations, while for the bi- and tri-confluent Heun equations the recurrence relations in general involve five terms. Other expansions for which the expansion coefficients obey recurrence relations involving more terms are also possible. The particular cases when these relations reduce to ones involving less number of terms are identified. The conditions for deriving closed-form finite-sum solutions via right-hand side termination of the constructed series are discussed.