Résumé
Liouville field theory on a sphere is considered. We explicitly derive adifferential equation for four-point correlation functions with one degeneratefield $V_{-\frac{mb}{2}}$. We introduce and study also a class of four-pointconformal blocks which can be calculated exactly and represented by finitedimensional integrals of elliptic theta-functions for arbitrary intermediatedimension. We study also the bootstrap equations for these conformal blocks andderive integral representations for corresponding four-point correlationfunctions. A relation between the one-point correlation function of a primaryfield on a torus and a special four-point correlation function on a sphere isproposed.