Remarks on irregular conformal blocks and Painlev\'e III and II tau functions

Understand

We prove a conjecture on uniqueness and existence of the irregular vertex operators of rank $r$ introduced in our previous paper.

• We also introduce ramified irregular vertex operators of the Virasoro algebra.
• As applications, we give conjectural formulas for series expansions of Painlev\'e III and II tau functions in terms of our ramified irregular conformal blocks.