The aim of this paper is to generalize the classical formula $e^xye^{-x}=sumlimits_{kge 0} frac{1}{k!} (ad~x)^k(y)$ by replacing $e^x$ with any formal power series $displaystyle {f(x)=1+sum_{kge 1} a_kx^k}$. We also obtain combinatorial applications to $q$-exponentials, $q$-binomials, and Hall-Littlewood polynomials.
We give an elementary proof of the Kontsevich conjecture that asserts that the iterations of the noncommutative rational map K_r:(x,y)-->(xyx^{-1},(1+y^r)x^{-1}) are given by noncommutative Laurent polynomials.
