Coefficients of multiplication formulas for classical orthogonal polynomials

Abstract

In this paper using both analytic and algorithmic approaches, we derive the coefficients \(D_m(n,a)\) of the multiplication formula $$\begin{aligned} p_n(ax)=\sum _{m=0}^nD_m(n,a)p_m(x) \end{aligned}$$ or the translation formula $$\begin{aligned} p_n(x+a)=\sum _{m=0}^nD_m(n,a)p_m(x), \end{aligned}$$ where \(\{p_n\}_{n\ge 0}\) is an orthogonal polynomial set, including the classical continuous orthogonal polynomials, the classical discrete orthogonal polynomials, the \(q\)-classical orthogonal polynomials, as well as the classical orthogonal polynomials on a quadratic lattice and a \(q\)-quadratic lattice. We give a representation of the coefficients \(D_m(n,a)\) as a single, double or triple sum whereas in many cases we get simple representations.