# 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.

## Key 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.

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Source: https://tomesphere.com/paper/1903.08162