Expansions of the solutions of the biconfluent Heun equation in terms of incomplete Beta and Gamma functions

TL;DR

This paper develops new series expansions for solutions of the biconfluent Heun equation using incomplete Beta and Gamma functions, providing a novel approach to solving this complex differential equation.

Contribution

The paper introduces two novel expansions of the biconfluent Heun solutions in terms of incomplete Beta functions, with recurrence relations for coefficients and potential for other special functions.

Findings

Two types of Beta function expansions derived

Recurrence relations for expansion coefficients established

Examples of Gamma function expansions provided

Abstract

Starting from equations obeyed by functions involving the first or the second derivatives of the biconfluent Heun function, we construct two expansions of the solutions of the biconfluent Heun equation in terms of incomplete Beta functions. The first series applies single Beta functions as expansion functions, while the second one involves a combination of two Beta functions. The coefficients of expansions obey four- and five-term recurrence relations, respectively. It is shown that the proposed technique is potent to produce series solutions in terms of other special functions. Two examples of such expansions in terms of the incomplete Gamma functions are presented