Identities for generalized Appell functions and the blow-up formula

In this paper, we prove identities for a class of generalized Appell functions which are based on the $\operatorname{A}_2$ root lattice. The identities are reminiscent of periodicity relations for the classical Appell function, and are proven using only analytic properties of the functions. Moreover they are a consequence of the blow-up formula for generating functions of invariants of moduli spaces of semi-stable sheaves of rank 3 on rational surfaces. Our proof confirms that in the latter context, different routes to compute the generating function (using the blow-up formula and wall-crossing) do arrive at identical $q$-series. The proof also gives a clear procedure for how to prove analogous identities for generalized Appell functions appearing in generating functions for sheaves with rank $r>3$.