On the vanishing of coefficients of the powers of a theta function

Abstract

A result on the Galois theory of q-difference equations (Sauloy in Théorie analytique locale des équations aux q-différences de pentes arbitraires. See arXiv:2006.03237v1, 2020) leads to the following question: If \(q \in {{\mathbf {C}}^*}\), \(\left| q \right| < 1\) and if one sets \({\theta _q}(z) := \sum \nolimits _{m \in {{\mathbf {Z}}}} q^{m(m-1)/2} z^m\), can some coefficients of the Laurent series expansion of \(\theta _q^k(z)\), \(k \in {{\mathbf {N}}}^*\), vanish ? We give a partial answer.

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Notes

  1. 1.

    Note however that the conventions in the present work are different, one assumes that \(0< \left| q \right| < 1\) and it is the formula (1) herebelow which defines \({\theta _q}\). Also, notation for the coefficients will differ, see formula (2).

  2. 2.

    This definition requires an unambiguous determination of \(q^{1/2}\), the necessary conventions are explained in 2.2.1.

  3. 3.

    Note however that in Sect. 3.1.4 we apply modular properties of generating series of number of representations by quadratic forms to obtain some complementary information (Proposition 7).

References

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    Berndt, B.C.: Ramanujan’s Notebooks. Part III. Springer, New York (1991)

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    Rademacher, H.: Topics in Analytic Number Theory, vol. 169. Springer, Berlin (1973)

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    Sauloy, J.: Théorie analytique locale des équations aux q-différences de pentes arbitraires (2020). Submitted for publication; see arXiv:2006.03237v1

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    Serre, J.-P.: A Course in Arithmetic. Translation of “Cours d’arithmetique”. 2nd corr. print., vol. 7. Springer, New York (1978)

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    Zhang, C.: A discrete summation for linear \(q\)-difference equations with analytic coefficients: general theory and examples. (Une sommation discréte pour des équations aux q-différences linéaires et á coefficients analytiques: Théorie générale et exemples.). In: Braaksma, B.L.J., et al. (eds.) Differential equations and the Stokes phenomenon. Proceedings of the conference, Groningen, Netherlands, May 28–30, 2001, pp. 309–329. World Scientific, Singapore (2002)

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Acknowledgements

Jacques Sauloy is indebted to Professor Berndt for suggesting to look at Ramanujan’s Notebooks and positive comments on first attempts.

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Correspondence to Jacques Sauloy.

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The work of Changgui Zhang was supported by Labex CEMPI (Centre Européen pour les Mathématiques, la Physique et leurs Interaction)

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Sauloy, J., Zhang, C. On the vanishing of coefficients of the powers of a theta function. Ramanujan J (2021). https://doi.org/10.1007/s11139-020-00366-8

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Keywords

  • Special functions
  • Theta functions
  • q-Calculus

Mathematics Subject Classification

  • 33E30
  • 39A13