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

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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.

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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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  1. Department of Mathematics, University of California, Davis, One Shields Ave., Davis, CA, 95616, USA

    Dan Romik

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  1. Dan Romik
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Correspondence to Dan Romik.

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This material is based upon work supported by the National Science Foundation under Grant No. DMS-1800725.

Appendix: a table of the values \((d(n))_{n=0}^{20}\)

Appendix: a table of the values \((d(n))_{n=0}^{20}\)

Table 1 The initial values of the sequence \((d(n))_{n=0}^\infty \)
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Romik, D. The Taylor coefficients of the Jacobi theta constant \(\theta _3\). Ramanujan J 52, 275–290 (2020). https://doi.org/10.1007/s11139-018-0109-5

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