  We recently proved that, when integrating out spacetime dependence with a
certain measure, four-point correlators $\langle
\mathcal{O}_2\mathcal{O}_2\mathcal{O}^{(i)}_p \mathcal{O}^{(j)}_p \rangle$ in
$SU(N)$ $\mathcal{N}=4$ super Yang-Mills are governed by a universal
Laplace-difference equation. Here $\mathcal{O}^{(i)}_p$ is a superconformal
primary with charge $p$ and degeneracy $i$. These observables, called
integrated correlators, are modular functions of coupling $\tau$. The
Laplace-difference equation relates integrated correlators of different charges
recursively. In this paper, we introduce generating functions for the
integrated correlators that sum over the charge. By utilising the
Laplace-difference equation, we determine the generating functions given
initial data. We show that the transseries of the integrated correlators in the
large-$p$ (large-charge) expansion consists of three parts: 1) is independent
of $\tau$ as a power series in $1/p$, plus an additional $\log(p)$ term if
$i=j$; 2) is a power series in $1/p$, with coefficients given by a sum of the
non-holomorphic Eisenstein series; 3) is a sum of exponentially decayed modular
functions in the large-$p$ limit, which can be viewed as a generalisation of
the non-holomorphic Eisenstein series. When $i=j$, there is an additional
modular function that is independent of $p$ and is determined by the integrated
correlator with $p=2$. The Laplace-difference equation was obtained with a
reorganisation of the operators that means the large-charge limit is taken in a
particular way here. From the $SL(2,\mathbb{Z})$-invariant results, we also
determine the generalised 't Hooft genus expansion and associated large-$p$
non-perturbative corrections of the integrated correlators by introducing
$\lambda = pg^2_{YM}$. The generating functions have subtle differences between
even and odd $N$ with important consequences in resurgence.
