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An elementary proof of the irrationality of Tschakaloff series

Abstract

We present a new proof of the irrationality of values of the series \(\mathcal{T}_q (z) = \sum\limits_{n = 0}^\infty {z^n q^{ - n(n - 1)/2} } \) in both qualitative and quantitative forms. The proof is based on a hypergeometric construction of rational approximations to T q (z).

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Correspondence to W. Zudilin.

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Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 6, pp. 59–64, 2005.

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Zudilin, W. An elementary proof of the irrationality of Tschakaloff series. J Math Sci 146, 5669–5673 (2007). https://doi.org/10.1007/s10958-007-0382-0

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Keywords

  • Rational Approximation
  • Elementary Proof
  • Golden Section
  • Theta Series
  • Analytic Number Theory