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
This definition requires an unambiguous determination of \(q^{1/2}\), the necessary conventions are explained in 2.2.1.
References
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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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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 57, 1125–1167 (2022). https://doi.org/10.1007/s11139-020-00366-8
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DOI: https://doi.org/10.1007/s11139-020-00366-8