Orthogonality via Transforms
Jim Freeman · Studies in Applied Mathematics · 1987
Let e(x, t) = ∑pn(x)tn be the generating function of a polynomial sequence, and the transform of multiplication by x relative to e(x, t). We show that the sequence pn(x) is orthogonal precisely when is a t‐variable, i.e., maps K[t] into itself and increases degree by 1. We also show how transform techniques can shed light on the recursion relations and differential equations for pn(x).