On fusion kernel in Liouville theory

We study fusion kernel for non-degenerate conformal blocks in Liouville theory as a solution to the difference equations originating from the pentagon identity. We suggest an approach to these equations based on 'non-perturbative' series expansion which allows to calculate the fusion kernel iteratively. We also find the exact solutions for the cases when the central charge is $c=1+6(b-b^{-1})^2$ and $b~\in \mathbb{N}$. For $c = 1$ our result reproduces the formula, obtained earlier from analytical continuation via Painlev\'e equation. However, in our case it appears in a significantly simplified form.