Identities for generalized Appell functions and the blow-up formula

TL;DR

This paper establishes identities for generalized Appell functions related to the A2 lattice, linking them to the blow-up formula for moduli space invariants, and confirms their consistency across different computational approaches.

Contribution

It proves new identities for generalized Appell functions based on the A2 lattice and connects these identities to the blow-up formula in algebraic geometry.

Findings

Identities for generalized Appell functions are proven using analytic properties.

The identities confirm the equivalence of different computational methods for generating functions.

Procedures are provided for extending identities to higher rank sheaves.

Abstract

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 $r>3$.