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On the relationship between modular and hypergeometric functions

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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.

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References

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Authors and Affiliations

  1. National Institute for Higher Education, Minsk, Belarus

    S. M. Artem’eva & H. -W. Burmann

  2. Mathematisches Institut, Georg-August-Universität Göttingen, Göttingen, Germany

    S. M. Artem’eva & H. -W. Burmann

Authors
  1. S. M. Artem’eva
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  2. H. -W. Burmann
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Additional information

Original Russian Text © S.M. Artem’eva, H.-W. Burmann, 2009, published in Differentsial’nye Uravneniya, 2009, Vol. 45, No. 2, pp. 147–153.

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Artem’eva, S.M., Burmann, H.W. On the relationship between modular and hypergeometric functions. Diff Equat 45, 151–158 (2009). https://doi.org/10.1134/S0012266109020013

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  • DOI: https://doi.org/10.1134/S0012266109020013

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