Summary: Riordan group concepts are combined with the basic properties of convolution families of polynomials and Sheffer sequences, to establish a duality law, canonical forms $\rho(n,m)={n\choose m}c^mF_{n-m}(m),\ c\neq0,$ and extensions $\rho(x,x-k)=(-1)^kx^{\underline{k+1}}c^{x-k}F_k(x)$, where the $F_k(x)$ are polynomials in $x$, holding for each $\rho(n,m)$ in a Riordan array. Examples $\rho(n,m)={n\choose m}S_k(x)$ are given, in which the $S_k(x)$ are "orthogonal" polynomials currently found in mathematical physics and combinatorial analysis.

05A19, 05A15

riordan group, convolution polynomials, polynomial extensions, Sheffer sequences, orthogonal polynomials