  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.
