Solutions of a Confluent Modification of the General Heun Equation in Terms of Generalized Hypergeometric Functions

We consider a confluent modification of the general Heun equation having three regular singular points and an irregular singularity of rank one at infinity. We show that this equation admits infinitely many solutions in terms of a single generalized confluent hypergeometric function. The solutions apply if two regular singularities have non-zero integer characteristic exponents and each of the two accessory parameters obeys a polynomial equation.