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General Eulerian Polynomials as Moments Using Exponential Riordan Arrays
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Paul Barry

School of Science

Waterford Institute of Technology

Ireland

**Abstract:**

Using the theory of exponential Riordan arrays and orthogonal
polynomials, we demonstrate that the general Eulerian polynomials, as
defined by Xiong, Tsao and Hall, are moment sequences for simple
families of orthogonal polynomials, which we characterize in terms of
their three-term recurrence. We obtain the generating functions of this
polynomial sequence in terms of continued fractions, and we also
calculate the Hankel transforms of the polynomial sequence. We indicate
that the polynomial sequence can be characterized by the further notion
of generalized Eulerian distribution first introduced by Morisita. We
finish with examples of related Pascal-like triangles.

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(Concerned with sequences
A000165
A000354
A001710
A007047
A007318
A008290
A046802
A060187
A080253.)

Received September 26 2013;
revised version received October 13 2013.
Published in *Journal of Integer Sequences*, November 16 2013.

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