Generalized-hypergeometric solutions of the biconfluent Heun equation

TL;DR

This paper explores power-series and Hermite function series solutions to the biconfluent Heun equation, identifying cases where solutions are expressed as linear combinations of generalized hypergeometric functions, expanding understanding of its solution space.

Contribution

It introduces new solution forms for the biconfluent Heun equation using generalized hypergeometric functions, including cases not reducible to polynomials.

Findings

Multiple cases where solutions are irreducible linear combinations of four hypergeometric functions

Solutions generally do not reduce to polynomials

Enhanced understanding of the solution structure of the biconfluent Heun equation

Abstract

We examine the power-series solutions and the series solutions in terms of the Hermite functions for the biconfluent Heun equation. Infinitely many cases for which a solution of the biconfluent equation is presented as an irreducible linear combination of four generalized hypergeometric functions, that in general do not reduce to polynomials, are identified.