Riordan matrix methods and properties of generating functions are used to prove that the entries of two Catalan-type Riordan arrays are linked to the Chebyshev polynomials of the first kind. The connections are that the rows of the arrays are used to expand the monomials (1/2) (2x) and (1/2) (4x) in terms of certain Chebyshev polynomials of degree n. In addition, we find new integral representa...

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 alternati...

Here presented a generalization of Catalan numbers and Catalan triangles associated with two parameters based on the sequence characterization of Bell-type Riordan arrays. Among the generalized Catalan numbers, a class of large generalized Catalan numbers and a class of small generalized Catalan numbers are defined, which can be considered as an extension of large Schröder numbers and small Sch...