Confluent hypergeometric expansions of the confluent Heun function governed by two-term recurrence relations

TL;DR

This paper demonstrates that for specific parameter choices, the confluent Heun function's solutions can be expanded into confluent hypergeometric functions with simplified two-term recurrence relations, allowing explicit coefficient calculation.

Contribution

It identifies infinite parameter sets where the three-term recurrence reduces to two-term relations and provides explicit formulas for the expansion coefficients.

Findings

Infinite parameter choices reduce recurrence relations to two-term forms

Explicit formulas for expansion coefficients are derived

Simplifies analysis of confluent Heun functions in special cases

Abstract

We show that there exist infinitely many nontrivial choices of parameters of the single confluent Heun equation for which the three-term recurrence relations governing the expansions of the solutions in terms of the confluent hypergeometric functions 1F1 and 0F1 are reduced to two-term ones. In such cases the expansion coefficients are explicitly calculated in terms of the Euler gamma functions.