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

George E. Andrews; Simon CF Rose

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arXiv · 1010.5769

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

George E. Andrews·Simon CF Rose

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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.

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George E. Andrews, Simon CF Rose. 2010-10-27. MacMahon's sum-of-divisors functions, Chebyshev polynomials, and Quasi-modular forms. https://arxiv.org/abs/1010.5769

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