Polynomial expansions and generating functions

AbstractWe study expansions in polynomials {Pn(x)}∞o generated by ∑∞n = o Pn(x)tn = A(t) φ(xtkθ(t)), θ(0) ≠ 0, and ∑∞n = 0 Pn(x)tn = ∑kj = 1 Aj(t) φ(xtϵj), ϵ1,…,ϵk being the k roots of unity. The case k = 1 is contained in a recent work by Fields and Ismail. We also prove a new generalization of Vandermond's inverse relations