Meixner-type results for Riordan arrays and associated integer sequences Paul Barry School of Science Waterford Institute of Technology Ireland pbarry@wit.ie Aoife Hennessy Department of Computing, Mathematics and Physics Waterford Institute of Technology Ireland aoife.hennessy@gmail.com Abstract We determine which (ordinary) Riordan arrays are the coefficient arrays of a family of orthogonal polynomials. In so doing, we are led to introduce a family of polynomi- als, which includes the Boubaker polynomials, and a scaled version of the Chebyshev poynomials, using the techniques of Riordan arrays. We classify these polynomials in terms of the Chebyshev polynomials of the first and second kinds. We also examine the Hankel transforms of sequences associated to the inverse of the polynomial coefficient arrays, including the associated moment sequences. 1 Introduction To each Riordan array (A(t),B(t)) we can associate a family of polynomials [19] by ∞ ∑ n=0 p n (x)t n =(A(t),B(t)) · 1 1 − xt = A(t) 1 − xB(t) . The question can then be asked as to what conditions must be satisfied by A(t) and B(t) in order to ensure that (p n (x)) n≥0 be a family of orthogonal polynomials. This can be considered to be a Meixner-type question [22], where the original Meixner result is related to Sheffer sequences (i.e., to exponential generating functions, rather than ordinary generating functions): ∞ ∑ n=0 p n (x)t n = A(t) exp(xB(t)). In providing an answer to this question, we shall introduce a two-parameter family of orthog- onal polynomials using Riordan arrays. These polynomials are inspired by the well-known 1