Some observations about determinants which are connected with Catalan numbers and related topics

TL;DR

This paper explores matrices with determinants equal to Catalan numbers, providing new proofs, generalizations, q-analogues, and connections to Hankel determinants, along with conjectures for future research.

Contribution

It introduces new methods and generalizations related to determinants connected with Catalan numbers, including q-analogues and Hankel determinant connections.

Findings

Matrices with Catalan number determinants analyzed

New proofs and generalizations presented

Connections to Hankel determinants and conjectures proposed

Abstract

In this (mostly expository) paper I want to share some observations prompted by a class of matrices whose determinants are Catalan numbers. Considering different methods of proof we obtain some generalizations and q-analogues and connections with Hankel determinants. Finally we state some conjectures.