A note on rank 5/2 Liouville irregular block, Painlev\'e 1 and the ${\cal H}_0$ Argyres-Douglas theory

We study 4d type ${\cal H}_0$ Argyres-Douglas theory in $\Omega$-background by constructing Liouville irregular state of rank 5/2. The results are compared with generalized Holomorphic anomaly approach, which provides order by order expansion in $\Omega$-background parameters $\epsilon_{1,2}$. Another crucial test of our results provides comparison with respect to Painlev\'{e} 1 $\tau$-function, which was expected to be hold in self-dual case $\epsilon_1=-\epsilon_2$. We also discuss Nekrasov-Shatashvili limit $\epsilon_1=0$, accessible either by means of deformed Seiberg-Witten curve, or WKB methods.