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A Note on a Family of Generalized Pascal Matrices Defined by Riordan Arrays
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Paul Barry

School of Science

Waterford Institute of Technology

Ireland

**Abstract:**

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.

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(Concerned with sequences
A000007
A000045
A000108
A000984
A007318
A029653
A094385
A110522
A114188
A114496
A156886
A156887
A157491.)

Received October 10 2012;
revised version received May 3 2013.
Published in *Journal of Integer Sequences*, May 16 2013.

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