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The Taylor coefficients of the Jacobi theta constant \(\theta _3\)

  • Dan RomikEmail author
Article
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Abstract

We study the Taylor expansion around the point \(x=1\) of a classical modular form, the Jacobi theta constant \(\theta _3\). This leads naturally to a new sequence \((d(n))_{n=0}^\infty =1,1,-1,51,849,-26199,\ldots \) of integers, which arise as the Taylor coefficients in the expansion of a related “centered” version of \(\theta _3\). We prove several results about the numbers d(n) and conjecture that they satisfy the congruence \(d(n)\equiv (-1)^{n-1}\ (\text {mod }5)\) and other similar congruence relations.

Keywords

Theta function Jacobi theta constant Modular form 

Mathematics Subject Classification

11F37 14K25 30B10 

Notes

Acknowledgements

The author thanks Robert Scherer, Christian Krattenthaler, Tanguy Rivoal, David Broadhurst, Peter Paule, Yiangjie Ye, Doron Zeilberger, Craig Tracy, David Bailey, and Bill Gosper for helpful discussions during the preparation of this manuscript. Some of these discussions took place during the author’s visit to the Erwin Schrödinger Institute (ESI) in November 2017; the author is grateful to ESI for its support and hospitality. The author also thanks the anonymous referee for suggesting useful corrections.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of California, DavisDavisUSA

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