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The zeros of the Weierstrass \(\wp\)–function and hypergeometric series

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Abstract

We express the zeros of the Weierstrass \(\wp\)-function in terms of generalized hypergeometric functions. As an application of our main result we prove the transcendence of two specific hypergeometric functions at algebraic arguments in the unit disc. We also give a Saalschützian 4 F 3–evaluation.

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References

  1. Archinard N. (2003). Exceptional sets of hypergeometric series. J. Number Theory 101: 244–269

    Article  MATH  MathSciNet  Google Scholar 

  2. Babister A.W. (1967). Transcendental Functions Satisfying Nonhomogeneous Linear Differential Equations. Macmillan, New York

    MATH  Google Scholar 

  3. Bailey W.N. (1964). Generalized hypergeometric series. Cambridge Tracts in Mathematics and Mathematical Physics. Stechert-Hafner Inc., New York

    Google Scholar 

  4. Baker A. (1990). Transcendental Number Theory, 2nd edn. Cambridge University Press, Cambridge

    Google Scholar 

  5. Beukers F. and Wolfart J. (1988). Algebraic Values of Hypergeometric Functions New Advances in Transcendence Theory (Durham, 1986). Cambridge University Press, Cambridge, 68–81

    Google Scholar 

  6. Eichler M. and Zagier D. (1981). On the zeros of the Weierstrass \({\wp}\)-function. Math. Ann. 258: 399–407

    Article  MathSciNet  Google Scholar 

  7. Eichler M. and Zagier D. (1985). The theory of Jacobi forms. Progress in Mathematics, vol 55. Birkhuser Boston Inc., Boston

    Google Scholar 

  8. Evans R.J. and Stanton D. (1984). Asymptotic formulas for zero-balanced hypergeometric series. SIAM J. Math. Anal. 15(5): 1010–1020

    Article  MATH  MathSciNet  Google Scholar 

  9. Fricke R. (1916). Die Elliptischen Funktionen und ihre Anwendungen I, II. B.G. Teubner, Leipzig

    Google Scholar 

  10. Nesterenko Yu. V. (2002). On the algebraic independence of numbers. A Panorama of Number Theory or the View from Baker’s garden (Zürich, 1999). Cambridge University Press, Cambridge, 148–167

    Google Scholar 

  11. Ramanujan, S.: Collected papers, (AMS Chelsea Publishing 2000) pp. 136–162

  12. Ramanujan S. (1957). Notebooks, vol. 2. Tata Institute of Fundamental Research, Bombay

    Google Scholar 

  13. Schneider T. (1937). Arithmetische Untersuchungen elliptischer Integrale. Math. Ann. 113: 1–13

    Article  MathSciNet  Google Scholar 

  14. Schwarz, H.A.: Formeln und Lehrsätze zum Gebrauche der elliptischen Functionen. Nach Vorlesungen und Aufzeichnungen des Herrn Professor K. Weierstrass. Göttingen. Dieterich. Bogen 11 u. 12, S. 81–96 (1885)

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Authors and Affiliations

  1. Mathematics Department, UCLA, Box 951555, Los Angeles, CA, 90095, USA

    W. Duke

  2. Mathematics Department, ETH, 8092, Zurich, Switzerland

    Ö. Imamoḡlu

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  1. W. Duke
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  2. Ö. Imamoḡlu
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Correspondence to Ö. Imamoḡlu.

Additional information

Research of W. Duke was supported in part by NSF Grant DMS-0355564. He wishes to acknowledge and thank the Forschungsinstitut für Mathematik of ETH Zürich for its hospitality and support.

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Duke, W., Imamoḡlu, Ö. The zeros of the Weierstrass \(\wp\)–function and hypergeometric series. Math. Ann. 340, 897–905 (2008). https://doi.org/10.1007/s00208-007-0174-3

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  • DOI: https://doi.org/10.1007/s00208-007-0174-3

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