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

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.