  We derive explicit expressions for the generating functions of refined
Vafa-Witten invariants $\Omega(\gamma,y)$ of $\mathbb{P}^2$ of arbitrary rank
$N$ and for their non-holomorphic modular completions. In the course of
derivation we also provide: i) a generalization of the recently found
generating functions of $\Omega(\gamma,y)$ and their completions for Hirzebruch
and del Pezzo surfaces in the canonical chamber of the moduli space to a
generic chamber; ii) a version of the blow-up formula expressed directly in
terms of these generating functions and its reformulation in a manifestly
modular form.
