Generalized confluent hypergeometric solutions of the Heun confluent equation

TL;DR

This paper demonstrates that the Heun confluent equation has infinitely many solutions expressed through confluent hypergeometric functions, with specific conditions on the characteristic exponents and accessory parameters, expanding the solution space.

Contribution

It introduces a new class of solutions for the Heun confluent equation using confluent hypergeometric functions, detailing conditions on exponents and parameters.

Findings

Solutions expressed as linear combinations of Kummer or Bessel functions

Characteristic exponents are non-zero integers

Accessory parameters satisfy polynomial equations

Abstract

We show that the Heun confluent equation admits infinitely many solutions in terms of the confluent generalized hypergeometric functions. For each of these solutions a characteristic exponent of a regular singularity of the Heun confluent equation is a non-zero integer and the accessory parameter obeys a polynomial equation. Each of the solutions can be written as a linear combination with constant coefficients of a finite number of either the Kummer confluent hypergeometric functions or the Bessel functions.