Abstract
We show that the eighth power of the Jacobi triple product is a Jacobi–Eisenstein series of weight 4 and index 4 and we calculate its Fourier coefficients. As applications we obtain explicit formulas for the eighth powers of theta-constants of arbitrary order and the Fourier coefficients of the Ramanujan \({\Delta }\)-function \({\Delta }(\tau )=\eta ^{24}(\tau ),\eta ^{12}(\tau )\) and \(\eta ^{8}(\tau )\) in terms of Cohen’s numbers H(3, N) and H(5, N). We give new formulas for the number of representations of integers as sums of eight higher figurate numbers. We also calculate the sixteenth and the twenty-fourth powers of the Jacobi theta-series using the basic Jacobi forms.
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Acknowledgements
The subject of this work is closely related to the course of lectures “Jacobi modular forms: 30 ans après” (see [8]) given by the first author in the Laboratory of Algebraic Geometry of the Department of Pure Mathematics of HSE in Moscow in 2015–2016. In this course, an explicit formula for the eighth power of the Jacobi triple product was mentioned as a possible application. We express our gratitude to all participants of the course, and especially to Dimitry Adler who was a research assistant of the course, and Guillaume Bioche who prepared a tex file of the lectures. The authors are grateful to reviewers for many useful comments.
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The study has been funded by the Russian Academic Excellence Project ‘5-100’. The second author was supported by LabEx CEMPI, Lille.
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Gritsenko, V., Wang, H. Powers of Jacobi triple product, Cohen’s numbers and the Ramanujan \({\Delta }\)-function. European Journal of Mathematics 4, 561–584 (2018). https://doi.org/10.1007/s40879-017-0185-x
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DOI: https://doi.org/10.1007/s40879-017-0185-x