Virasoro conformal blocks in closed form

Virasoro conformal blocks are fixed in principle by symmetry, but a closed-form expression is unknown in the general case. In this work, we provide two new closed-form expansions for the four-point Virasoro blocks on the sphere, for arbitrary operator dimensions and central charge c. We do so by solving known recursion relations. One representation is a surn over hypergeomaric global blocks, whose coefficients we provide at arbitrary level. The other is a sum over semiclassical Virasoro blocks obtained in the limit in which two external operator dimensions scale linearly with large c. In both cases, the expansion of the Virasoro blocks is easily extracted. We discuss applications of these expansions to entanglement and thermality in conformal field theories and particle scattering in three-dimensional quantum gravity.