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 ρ(n,m) = ( n m ) cFn−m(m), c 6= 0, and extensions ρ(x, x − k) = (−1) xcFk(x), where the Fk(x) are polynomials in x, holding for each ρ(n,m) in a Riordan array. Examples ρ(n,m) = ( n m ) Sk(x) are given, in which the Sk(x) are “orthogonal” polynomials currently found in mathematical physics and combinatorial analysis.