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

Sheaves on P2 and generalized Appell functions