  A closed expression is given for the generating function of (virtual)
Poincar\'e polynomials of moduli spaces of semi-stable sheaves on the
projective plane $\mathbb{P}^2$ with arbitrary rank $r$ and Chern classes. This
generating function is known to equal the partition function of topologically
twisted gauge theory with $\mathcal{N}=4$ supersymmetry and gauge group $U(r)$,
which localizes on the Hermitian Yang-Mills solutions of the gauge field. To
classify and study the novel generating functions, the notion of Appell
functions with signature $(n_+,n_-)$ is introduced. For $n_-=1$, these novel
functions reduce to the known class of Appell functions with multiple variables
or higher level.
