Orthogonality via Transforms

Let e ( x, t ) = ∑p n ( x ) t n 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 p n ( 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 p n ( x ).