q-Catalan bases and their dual coefficients

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.