Catalan Numbers and Jacobi Polynomials

TL;DR

This paper proves that the inverse of a Hankel matrix formed from reciprocals of Catalan numbers has integer entries, extending to generalized Catalan numbers and connecting to Jacobi polynomial variants.

Contribution

It establishes the integrality of the inverse Hankel matrix entries for Catalan and generalized Catalan numbers using orthogonal polynomials and computer algebra techniques.

Findings

Inverse Hankel matrix entries are integers for Catalan numbers

Generalization to an infinite family of generalized Catalan numbers

Use of orthogonal polynomials and Wilf-Zeilberger theory in proofs

Abstract

We prove that the inverse of the Hankel matrix of the reciprocals of the Catalan numbers has integer entries. We generalize the result to an infinite family of generalized Catalan numbers. The Hankel matrices that we consider are associated with orthogonal polynomials that are variants of Jacobi polynomials. Our proofs use these polynomials and computer algebra based on Wilf-Zeilberger theory.