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On the relationship between modular and hypergeometric functions

On the relationship between modular and hypergeometric functions We study the relationship between two Hecke theta series, the Dedekind function, and the Gauss hypergeometric function. The main result of the present paper is given by formulas for the representation of the theta series in the form of compositions of the squared Dedekind function, a power of the absolute invariant, and canonical integrals of the second-order hypergeometric differential equation with special values of the three parameters. The proofs of these representations are based on the properties of the matrix transforming the canonical integrals of the Gauss equation in a neighborhood of zero into canonical integrals of the same equation in a neighborhood of unity. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Differential Equations Springer Journals

On the relationship between modular and hypergeometric functions

Differential Equations , Volume 45 (2) – Mar 31, 2009

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References (7)

Publisher
Springer Journals
Copyright
Copyright © 2009 by Pleiades Publishing, Ltd.
Subject
Mathematics; Difference and Functional Equations; Partial Differential Equations; Ordinary Differential Equations
ISSN
0012-2661
eISSN
1608-3083
DOI
10.1134/S0012266109020013
Publisher site
See Article on Publisher Site

Abstract

We study the relationship between two Hecke theta series, the Dedekind function, and the Gauss hypergeometric function. The main result of the present paper is given by formulas for the representation of the theta series in the form of compositions of the squared Dedekind function, a power of the absolute invariant, and canonical integrals of the second-order hypergeometric differential equation with special values of the three parameters. The proofs of these representations are based on the properties of the matrix transforming the canonical integrals of the Gauss equation in a neighborhood of zero into canonical integrals of the same equation in a neighborhood of unity.

Journal

Differential EquationsSpringer Journals

Published: Mar 31, 2009

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