Notes on the Hankel transform of linear combinations of consecutive pairs of Catalan numbers

TL;DR

This paper explores the Hankel transform of linear combinations of consecutive Catalan numbers, generalizing known formulas and revealing new number triangles analyzed via Riordan arrays.

Contribution

It introduces a generalized formula for the Hankel transform of linear combinations of Catalan numbers and connects it to Riordan array analysis.

Findings

New closed-form expressions for Hankel transforms

Identification of novel number triangles

Application of Riordan arrays to analyze these structures

Abstract

We provide a context around a conjectured closed form for the Hankel transform of linear combinations of consecutive pairs of Catalan numbers. This generalizes the formula for the Hankel transforms of the shifted Catalan numbers and the known results for linear combinations of pairs of Catalan numbers. Many interesting number triangles emerge, some of which we analyze using the language of Riordan arrays.