Riordan Arrays, Sheffer Sequences and "Orthogonal" Polynomials

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

are polynomials in

, 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

are ``orthogonal'' polynomials currently found in mathematical physics and combinatorial analysis.

Received October 17 2008; revised version received December 11 2008. Published in Journal of Integer Sequences, December 11 2008.

Riordan Arrays, Sheffer Sequences and "Orthogonal" Polynomials

Giacomo Della Riccia Dept. of Math. and Comp. Science - Research Center Norbert Wiener University of Udine Via delle Scienze 206 33100 Udine Italy

Giacomo Della Riccia
Dept. of Math. and Comp. Science - Research Center Norbert Wiener
University of Udine
Via delle Scienze 206
33100 Udine
Italy