Journal of Integer Sequences, Vol. 11 (2008), Article 08.5.3

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


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.

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

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