Degeneration limits of Virasoro vertex operators and Painlev\'e tau functions

We construct degeneration limits of vertex operators for the Virasoro algebra. Our method relies on the rearranged expansion of compositions of vertex operators together with their integral representations. Using this framework, we obtain a vertex operator between Verma modules of rank $r+1$ as a degeneration of a composition of two vertex operators between Verma modules of rank $r$ ($r\in\mathbb{Z}_{\geq 0}$). Furthermore, we apply these degeneration limits to prove the conjectural expansions of the $\tau$ functions of the fifth and fourth Painlev\'e equations in terms of irregular conformal blocks [H. Nagoya, J. Math. Phys. 56, 123505 (2015)].