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Generalized Narayana Polynomials, Riordan Arrays, and Lattice Paths
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Paul Barry

School of Science

Waterford Institute of Technology

Ireland

Aoife Hennessy

Department of Computing, Mathematics and Physics

Waterford Institute of Technology

Ireland

**Abstract:**

We study a family of polynomials in two variables,
identifying them as the moments of a two-parameter family of orthogonal
polynomials. The coefficient array of these orthogonal polynomials is
shown to be an ordinary Riordan array. We express the generating
function of the sequence of polynomials under study as a continued
fraction, and determine the corresponding Hankel transform. An
alternative characterization of the polynomials in terms of a related
Riordan array is also given. This Riordan array is associated with
Łukasiewicz paths. The special form of the production matrices is
exhibited in both cases. This allows us to produce a bijection from a
set of colored Łukasiewicz paths to a set of colored Motzkin paths.
The polynomials studied generalize the notion of Narayana polynomial.

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(Concerned with sequences
A000007
A000012
A000045
A000108
A000169
A000984
A001003
A001263
A001850
A006125
A007318
A007564
A008459
A059231
A064062
A064310
A069835
A083667
A084771
A099169
A101850
A143464
A155084
A187021.)

Received November 9 2011;
revised version received April 17 2012.
Published in *Journal of Integer Sequences*, April 20 2012.

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