q-power symmetric functions and q-exponential formula

Abstract

Let $\hat{I}»=\left(\hat{I}{»}_1,\hat{I}{»}_2,...,\hat{I}{»}_r\right)$ be an integer partition, and $\left[p_{\hat{I}»}\right]$ the $q$-analog of the symmetric power function . This $q$-analogue has been defined as a special case, in the author's previous article: "A $q$-analog of certain symmetric functions and one of its specializations". Here, we prove that a large part of the classical relations between $p_{\hat{I}»}$, on one hand, and the elementary and complete symmetric functions $e_n$ and $h_n$, on the other hand, have $q$-analogues with $\left[p_{\hat{I}»}\right]$. In particular, the generating functions $E\left(t\right)={\sum }_{n\ge 0}{e}_n{t}^n$ and $H\left(t\right)={\sum }_{n\ge 0}{h}_n{t}^n$ are expressed in terms of $\left[p_n\right]$, using Gessel's $q$-exponential formula and a variant of it. A factorization of these generating functions into infinite $q$-products, which has no classical counterpart, is established. By specializing these results, we show that the $q$-binomial theorem is a special case of these infinite $q$-products. We also obtain new formulas for the tree inversions enumerators and for certain $q$-orthogonal polynomials, detailing the case of dicrete $q$-Hermite polynomials.

23 pages. Correction of Equation (2.4)