Identities involving Narayana polynomials and Catalan numbers

TL;DR

This paper explores relationships between Narayana polynomials, Catalan numbers, and Legendre polynomials, providing multiple proofs and applications that yield new identities and deepen understanding of these combinatorial objects.

Contribution

It establishes integral representations of Narayana polynomials, expresses Catalan numbers via Narayana polynomials through three identities, and offers multiple proofs and applications.

Findings

Narayana polynomials can be represented as integrals of Legendre polynomials

Catalan numbers can be expressed in terms of Narayana polynomials through three identities

The paper provides algebraic and combinatorial proofs for these identities

Abstract

We first establish the result that the Narayana polynomials can be represented as the integrals of the Legendre polynomials. Then we represent the Catalan numbers in terms of the Narayana polynomials by three different identities. We give three different proofs for these identities, namely, two algebraic proofs and one combinatorial proof. Some applications are also given which lead to many known and new identities.