Generating functions via integral transforms (Q878488)

Let \(\{P_n\}_{n\geq0}\) be a polynomial sequence with complex coefficients and generated by \[ G(x,t)=\sum_{n=0}^{\infty}P_n(x)t^n. \] Starting from this generating function, the authors find other generating functions of the type: \[ G_\gamma(x,t)=\sum_{n=0}^{\infty}\gamma_n P_n(x)t^n, \] where \(\{\gamma_n\}_{n\geq0}\) is a real number sequence independent on \(x\) and \(t\) and having the integral representation: \[ \gamma_n=\int_{0}^{\infty}t^nd\mu(t),\quad n=0,1,\ldots \] \(\mu\) being a nonnegative measure on \([0,\infty[\). They treat separately the cases where the support is \([a,b]\subset [0,\infty[\) and the case where the support is \([0,\infty[\) with the additional condition: \(\lim_{n\longrightarrow +\infty}{\gamma_{n+1}\over \gamma_n}=+\infty\). It was shown that the derived results in this paper are an unification and a generalization of some known results obtained by Borel, Rainville, McBride and Srivastava. Various explicit examples were considered for illustration.