Resumen de Une fonction eulerienne formelle
G. Cauchon
In this paper, we state and prove a formal version of a factorisation theorem, a particular case of which, due to A. Unterberger [2], concerns the operator $p^2 + q^2 -\lambda^2$ where $\lambda\in ]1/2,+\infty[$ and $p, q$ are operators on the Poincaré half plane $\{z\in\mathbb{ C}\vert Re(z) > 0\}$ , such that $[p, q] = q$. \newline As this factorisation involves operators in the form $\Gamma(ap+b)$ (defined in the sens of the spectral theory), where $\Gamma$ is the well known Euler function and $(a, b)\in (\mathbb{C}\setminus \{0\})\times\mathbb{C}$, we construct a formal version for those operators, and explain why this construction cannot be done in terms of formal series in $p$.
