q-Catalan bases and their dual coefficients

TL;DR

This paper introduces q-Catalan bases, explores their dual coefficients, and connects these concepts to q-commuting equations and Garsia's work, revealing broader generalizations of q-Catalan number properties.

Contribution

It defines q-Catalan bases, analyzes their dual coefficients, and links these to solutions of q-commuting equations and existing theories, expanding understanding of q-Catalan number properties.

Findings

Dual coefficients appear in q-commuting equations

Solution of an abstract q-Segner recursion

Connections to Garsia's 1981 work

Abstract

We define q-Catalan bases which are a generalization of the q-polynomials z^n(z,q)_n. The determination of their dual bases involves some q-power series termed dual coefficients. We show how these dual coefficients occur in the solution of some equations with q-commuting coefficients and solve an abstract q-Segner recursion. We study the connection between this theory and Garsia's (1981). The overall flavor of this work is to show how some properties of q-Catalan numbers are in fact instances of much more general results on dual coefficients.