MacMahon's sum-of-divisors functions, Chebyshev polynomials, and Quasi-modular forms

TL;DR

This paper explores the connection between MacMahon's sum-of-divisors functions, Chebyshev polynomials, and quasi-modular forms, providing recurrence relations and proving a conjecture related to their structure, with applications to curve counting on Abelian surfaces.

Contribution

It establishes a new relationship between sum-of-divisors functions, Chebyshev polynomials, and quasi-modular forms, including a proof of MacMahon's conjecture and recurrence relations.

Findings

Derived recurrence relations for the functions.

Proved MacMahon's conjecture on their form.

Linked these functions to solutions of curve-counting problems.

Abstract

We investigate a relationship between MacMahon's generalized sum-of-divisors functions and Chebyshev polynomials of the first kind. This determines a recurrence relation to compute these functions, as well as proving a conjecture of MacMahon about their general form by relating them to quasi-modular forms. These functions arise as solutions to a curve-counting problem on Abelian surfaces.