Identities involving Narayana polynomials and Catalan numbers

Understand

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.