We prove a Fredholm determinant and short-distance series representation of the Painlevé V tau function τ (t ) associated to generic monodromy data. Using a relation of τ (t ) to two different types of irregular c = 1 Virasoro conformal blocks and the confluence from Painlevé VI equation, connection formulas between the parameters of asymptotic expansions at 0 and i ∞ are conjectured. Explicit evaluations of the connection constants relating the tau function asymptotics as t → 0, +∞, i ∞ are obtained. We also show that irregular conformal blocks of rank 1, for arbitrary central charge, are obtained as confluent limits of the regular conformal blocks.

We construct confluent conformal blocks of the second kind of the Virasoro algebra. We also construct the Stokes transformations which map such blocks in one Stokes sector to another. In the BPZ limit, we verify explicitly that the constructed blocks and the associated Stokes transformations reduce to solutions of the confluent BPZ equation and its Stokes matrices, respectively. Both the confluent conformal blocks and the Stokes transformations are constructed by taking suitable confluent limits of the crossing transformations of the four-point Virasoro conformal blocks.

We study the dependence of the tau function of Painlevé I equation on the generalized monodromy of the associated linear problem. In particular, we compute connection constants relating the tau function asymptotics on five canonical rays at infinity. The result is expressed in terms of dilogarithms of cluster type coordinates on the space of Stokes data.