A Note on a Family of Generalized Pascal Matrices Defined by Riordan Arrays
School of Science
Waterford Institute of Technology
We study the properties of a parameterized family of
generalized Pascal matrices, defined by Riordan arrays. In particular,
we characterize the central elements of these lower triangular
matrices, which are analogues of the central binomial coefficients. We
then specialize to the value 2 of the parameter, and study the
inverse of the matrix in question, and in particular we study the
sequences given by the first column and row sums of the inverse matrix.
Links to moments and orthogonal polynomials are examined, and Hankel
transforms are calculated. We study the effect of the powers of the
binomial matrix on the family. Finally we posit a conjecture concerning
determinants related to the Christoffel-Darboux bivariate quotients
defined by the polynomials whose coefficient arrays are given by the
generalized Pascal matrices.
Full version: pdf,
(Concerned with sequences
Received October 10 2012;
revised version received May 3 2013.
Published in Journal of Integer Sequences, May 16 2013.
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