Confluent hypergeometric expansions of the solutions of the double-confluent Heun equation

TL;DR

This paper explores new expansions of solutions to the double-confluent Heun equation using confluent hypergeometric functions, analyzing recurrence relations and conditions for finite solutions.

Contribution

It introduces novel hypergeometric expansions for the double-confluent Heun equation and examines recurrence relations and finite sum solutions.

Findings

Two sets of expansions have three-term recurrence relations.

A third expansion involves a five-term recurrence relation.

An example expansion with a seven-term recurrence relation is provided.

Abstract

Several expansions of the solutions of the double-confluent Heun equation in terms of the Kummer confluent hypergeometric functions are presented. Three different sets of these functions are examined. Discussing the expansions without a pre-factor, it is shown that two of these functions provide expansions the coefficients of which obey three-term recurrence relations, while for the third confluent hypergeometric function the corresponding recurrence relation generally involves five-terms. The latter relation is reduced to a three-term one only in the case when the double-confluent Heun equation degenerates to the confluent hypergeometric equation. The conditions for obtaining finite sum solutions via termination of the expansions are discussed. The possibility of constructing expansions of different structure using certain equations related to the double-confluent Heun equation is discussed. An example of such expansion derived using the equation obeyed by a function involving the derivative of a solution of the double-confluent Heun equation is presented. In this way, expansions governed by three- or more term recurrence relations for expansion coefficients can be constructed. An expansion with coefficients obeying a seven-term recurrence relation is presented. This relation is reduced to a five-term one if the additional singularity of the equation obeyed by the considered auxiliary function coincides with a singularity of the double-confluent Heun equation.