  The aim of this paper is to generalize the classical formula
$e^xye^{-x}=\sum\limits_{k\ge 0} \frac{1}{k!} (ad~x)^k(y)$ by replacing $e^x$
with any formal power series $\displaystyle {f(x)=1+\sum_{k\ge 1} a_kx^k}$. We
also obtain combinatorial applications to $q$-exponentials, $q$-binomials, and
Hall-Littlewood polynomials.
