Difference equation satisfied by the Stieltjes transform of a sequence.

We study a class of generating functions related to the Stieltjes transform of a sequence of moments with respect to the basis of falling factorial polynomials. Given a recurrence relation for the coefficient sequence, it is shown how to compute the difference equation satisified by its generating function w.r.t. this basis. We give several examples from the class of discrete semiclassical orthogonal polynomials.