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

George E. Andrews; Simon C.F. Rose

doi:10.1515/CRELLE.2011.179

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

George E. Andrews, Simon C.F. Rose

Mathematics

Research output: Contribution to journal › Article › peer-review

30 Scopus citations

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.

Original language	English (US)
Pages (from-to)	97-103
Number of pages	7
Journal	Journal fur die Reine und Angewandte Mathematik
Volume	2013
Issue number	676
DOIs	https://doi.org/10.1515/CRELLE.2011.179
State	Published - 2013

All Science Journal Classification (ASJC) codes

General Mathematics
Applied Mathematics

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10.1515/CRELLE.2011.179

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

Abstract

All Science Journal Classification (ASJC) codes

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