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MacMahon's sum-of-divisors functions, Chebyshev polynomials, and quasi-modular forms
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George E. Andrews
and
Simon C. F. Rose
Published/Copyright:
December 23, 2011
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.
Received: 2010-11-09
Revised: 2011-05-02
Published Online: 2011-12-23
Published in Print: 2013-03
©[2013] by Walter de Gruyter Berlin Boston
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- Extension of plurisubharmonic functions with growth control
- Additivity and non-additivity for perverse signatures
- MacMahon's sum-of-divisors functions, Chebyshev polynomials, and quasi-modular forms
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Articles in the same Issue
- Conjecture de type de Serre et formes compagnons pour GSp4
- Extension of plurisubharmonic functions with growth control
- Additivity and non-additivity for perverse signatures
- MacMahon's sum-of-divisors functions, Chebyshev polynomials, and quasi-modular forms
- The prime geodesic theorem
- Inequities in the Shanks–Rényi prime number race: An asymptotic formula for the densities
- Klein approximation and Hilbertian fields
- Ricci flow on asymptotically conical surfaces with nontrivial topology
