Moments of orthogonal polynomials and exponential generating functions

Starting from the moment sequences of classical orthogonal polynomials we derive the orthogonality purely algebraically. We consider also the moments of (q=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q=1$$\end{document}) classical orthogonal polynomials, and study those cases in which the exponential generating function has a nice form. In the opposite direction, we show that the generalized Dumont–Foata polynomials with six parameters are the moments of rescaled continuous dual Hahn polynomials. Finally, we show that one of our methods can be applied to deal with the moments of Askey–Wilson polynomials.