Generalized Narayana Polynomials, Riordan Arrays and Lattice Paths

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 Lukasiewicz 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 Lukasiewicz paths to a set of colored Motzkin paths. The polynomials studied generalize the notion of Narayana polynomial.