Coefficients of Multiplication Formulas for Classical Orthogonal Polynomials

In this paper, using both an analytic and algorithmic approach, we derive the coefficients Dm(n, a) of the multiplication formula pn(ax) = n ∑ m=0 Dm(n, a)pm(x) or the translation formula pn(x +a) = n ∑ m=0 Dm(n, a)pm(x), where {pn}n≥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 and a q-quadratic lattice. We give a representation of the coefficients Dm(n, a) as a single, double or triple sum whereas in many cases we get simple representations.