  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 $r>3$.
