Journal of Integer Sequences, Vol. 15 (2012), Article 12.4.8

Generalized Narayana Polynomials, Riordan Arrays, and Lattice Paths

Paul Barry
School of Science
Waterford Institute of Technology

Aoife Hennessy
Department of Computing, Mathematics and Physics
Waterford Institute of Technology


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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