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


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.

This is a preview of subscription content, access via your institution.


  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).


  1. 1.

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

    Google Scholar 

  2. 2.

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

    Google Scholar 

  3. 3.

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

  4. 4.

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

    Google Scholar 

  5. 5.

    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)

Download references


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

Author information



Corresponding author

Correspondence to Jacques Sauloy.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

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

Download citation


  • Special functions
  • Theta functions
  • q-Calculus

Mathematics Subject Classification

  • 33E30
  • 39A13