A study of polynomials associated with some generating functions

Many familiar polynomials such as the Hermite polynomials and the Appell polynomials have the generating functions of the form

F(x,t)=A(t)G(xt)=^∞_n=0∑g_n(x)tⁿ.

We say that F(x,t) generates the sequence of polynomials {g_n(x)} and that F(x,t) is a generating function for the g_n(x).

The purpose of this paper is to develop some properties of the polynomials g_n(x) associated with certain choices of A(t). These will include some recurrence relations which follow from a partial differential equation satisfied by the generating function F(x,t). Also, we study the expansion of a function in terms of these polynomials in both real and complex domains and give some conditions under which such an expansion converges.