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

  • Dan RomikEmail author


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.


Theta function Jacobi theta constant Modular form 

Mathematics Subject Classification

11F37 14K25 30B10 



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.


  1. 1.
    Bailey, W.N.: Generalized Hypergeometric Series. Cambridge University Press, Cambridge (1935)zbMATHGoogle Scholar
  2. 2.
    Berndt, B.C.: Ramanujan’s Notebooks, Part III. Springer, New York (1991)CrossRefGoogle Scholar
  3. 3.
    Berndt, B.C.: Ramanujan’s Notebooks, Part V. Springer, New York (1998)CrossRefGoogle Scholar
  4. 4.
    Datskovsky, B., Guerzhoy, P.: \(p\)-adic interpolation of Taylor coefficients of modular forms. Math. Ann. 340, 465–476 (2008)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Edwards, H.M.: Riemann’s Zeta Function. Dover Publications, New York (2001)zbMATHGoogle Scholar
  6. 6.
    Larson, H., Smith, G.: Computing properties of Taylor coefficients of modular forms. Int. J. Number Theory 10, 1501–1518 (2014)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Ohyama, Y.: Differential relations of theta functions. Osaka J. Math. 32, 431–450 (1995)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Shimura, G.: On the derivatives of theta functions and modular forms. Duke Math. J. 44, 365–387 (1977)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Stanley, R.P.: Enumerative Combinatorics, vol. 2. Cambridge University Press, Cambridge (1999)CrossRefGoogle Scholar
  10. 10.
    Villegas, F.R., Zagier, D.: Square roots of central values of Hecke \(L\)-series. In: Gouvea, F.Q., Yui, N. (eds.) Advances in Number Theory (Proceedings of the Third Conference of the Canadian Number Theory Association), pp. 81–99. Oxford University Press, Oxford (1993)Google Scholar
  11. 11.
    Villegas, F.R., Zagier, D.: Which primes are sums of two cubes? In: Dilcher, K. (ed.) Number Theory (Proceedings of the Fourth Conference of the Canadian Number Theory Association), pp. 295–306. CMS Conference Proceedings 15 (1995)Google Scholar
  12. 12.
    Voight, J., Willis, J.: Computing power series expansions of modular forms. In: öckle, G.B, Wiese, G. (eds.) Computations with Modular Forms (Proceedings of a Summer School and Conference, Heidelberg, August/September 2011). Contributions in Mathematical and Computational Sciences, vol. 6, pp. 331–361. Springer, New York (2014)Google Scholar
  13. 13.
    Whittaker, E.T., Watson, G.N.: A Course of Modern Analysis, 4th edn. Cambridge University Press, Cambridge (1996)CrossRefGoogle Scholar
  14. 14.
    Wünsche, A.: Generating functions for products of special Laguerre 2D polynomials and Hermite 2D polynomials. Appl. Math. 6, 2142–2168 (2015)CrossRefGoogle Scholar
  15. 15.
    Zagier, D.: Elliptic modular functions and their applications. In: K. Ranestad (ed.) The 1-2-3 of Modular Forms, pp. 1–103. Springer, New York (2008)Google Scholar
  16. 16.
    Zeilberger, D.: A user’s manual for the Maple program Theta3Romik.txt implementing Dan Romik’s article “The Taylor coefficients of the Jacobi \(\theta _3\) constant. Accessed 23 July 2018

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© 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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