q-power symmetric functions and q-exponential formula

Abstract

Let $\lambda =\left( \lambda_{1},\lambda_{2},...,\lambda_{r}\right) $ be an integer partition, and $\left[p_{\lambda }\right] $ the $q$-analog of the symmetric power function $%p_{\lambda }$. 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 $p_{\lambda }$, 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_{\lambda }\right] $. In particular, the generating functions $E\left( t\right) =\sum\nolimits_{n\geq 0}e_{n}t^{n}$ and $H\left( t\right) =\sum\nolimits_{n\geq 0}h_{n}t^{n}$ 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.