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On algebraic relations for Ramanujan’s functions

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Abstract

Let P,Q, and R denote the Ramanujan Eisenstein series. We compute algebraic relations in terms of P(q i) (i=1,2,3,4), Q(q i) (i=1,2,3), and R(q i) (i=1,2,3). For complex algebraic numbers q with 0<|q|<1 we prove the algebraic independence over ℚ of any three-element subset of {P(q),P(q 2),P(q 3),P(q 4)} and of any two-element subset of {Q(q),Q(q 2),Q(q 3)} and {R(q),R(q 2),R(q 3)}, respectively. For all the results we use some expressions of \(P(q^{i_{1}}), Q(q^{i_{2}}) \), and \(R(q^{i_{3}}) \) in terms of theta constants. Computer-assisted computations of functional determinants and resultants are essential parts of our proofs.

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Authors and Affiliations

  1. Fachhochschule für die Wirtschaft, University of Applied Sciences, Freundallee 15, 30173, Hannover, Germany

    Carsten Elsner

  2. Department of Mathematics, Keio University, Hiyoshi, Kohoku–ku, Yokohama, 223–8522, Japan

    Iekata Shiokawa

Authors
  1. Carsten Elsner
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  2. Iekata Shiokawa
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Correspondence to Carsten Elsner.

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Elsner, C., Shiokawa, I. On algebraic relations for Ramanujan’s functions. Ramanujan J 29, 273–294 (2012). https://doi.org/10.1007/s11139-012-9384-8

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  • DOI: https://doi.org/10.1007/s11139-012-9384-8

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