Geodesic
Riordan arrays, Sheffer sequences and ``orthogonal'' polynomials
Journal of integer sequences, Tome 11 (2008) no. 5
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.
Classification : 05A19, 05A15
Keywords: riordan group, convolution polynomials, polynomial extensions, Sheffer sequences, orthogonal polynomials 
@article{JIS_2008__11_5_a5,
     author = {Della Riccia,  Giacomo},
     title = {Riordan arrays, {Sheffer} sequences and ``orthogonal'' polynomials},
     journal = {Journal of integer sequences},
     year = {2008},
     volume = {11},
     number = {5},
     zbl = {1165.05311},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JIS_2008__11_5_a5/}
}
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Della Riccia,  Giacomo. Riordan arrays, Sheffer sequences and ``orthogonal'' polynomials. Journal of integer sequences, Tome 11 (2008) no. 5. http://geodesic.mathdoc.fr/item/JIS_2008__11_5_a5/