Incomplete beta-function expansions of the solutions to the confluent Heun equation

Abstract

Several expansions of the solutions to the confluent Heun equation in terms of incomplete Beta functions are constructed. A new type of expansion involving certain combinations of the incomplete Beta functions as expansion functions is introduced. The necessary and sufficient conditions when the derived expansions are terminated, thus generating closed-form solutions, are discussed. It is shown that termination of a Beta-function series solution always leads to a solution that is necessarily an elementary function.

References in corpus (2)

• Solutions for the General, Confluent and Biconfluent Heun equations and their connection with Abel equations
• New solutions of Heun general equation

Cited by in corpus (11)

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• Potentials of the Heun class
• Expansions of the solutions to the confluent Heun equation in terms of the Kummer confluent hypergeometric functions
• Expansions of the solutions of the confluent Heun equation in terms of the incomplete Beta and the Appell generalized hypergeometric functions
• Expansions of the solutions of the general Heun equation governed by two-term recurrence relations for coefficients
• Elementary Integral Series for Heun Functions. With an Application to Black-Hole Perturbation Theory
• Confluent hypergeometric expansions of the confluent Heun function governed by two-term recurrence relations
• On the Liouvillian solutions to the perturbation equations of the Schwarzschild black hole
• Series solutions of confluent Heun equations in terms of incomplete Gamma-functions