Remarks on some series expansions associated with certain products of the incomplete Gamma functions

In several recent works, some interesting generalizations of the first-order Volterra-type integro-differential equation governing the unsaturated behavior of the free electron laser (FEL) were introduced and investigated by making use of fractional calculus (that is, calculus of integrals and derivatives of an arbitrary real or complex order). Among other things, it is observed here that an expansion formula for the confluent hypergeometric function in a series of the product of two entire (integral) incomplete Gamma functions does not hold true as asserted and applied in these earlier works. Some necessary corrections and possible remedies are also pointed out.