On One-Parameter Catalan Arrays

TL;DR

This paper introduces a parametric family of Riordan arrays derived from Catalan triangles, exploring their properties and deriving combinatorial identities involving Chebyshev polynomials, Fibonacci numbers, and periodic sequences.

Contribution

It presents a new class of Riordan arrays formed by multiplying existing arrays with generalized Pascal arrays, focusing on one-parameter Catalan triangles and their combinatorial properties.

Findings

Derived combinatorial identities involving Catalan matrices, Chebyshev polynomials, Fibonacci numbers, and periodic sequences.

Established properties of the one-parameter Catalan triangles within the Riordan array framework.

Explored relationships between these arrays and classical combinatorial sequences.

Abstract

We present a parametric family of Riordan arrays which are obtained by multiplying any Riordan array with a generalized Pascal array. In particular, we focus on some interesting properties of one-parameter Catalan triangles. We obtain several combinatorial identities that involve two special Catalan matrices, the Chebyshev polynomials of the second kind, some periodic sequences, and the Fibonacci numbers.