Skip to main content

Automatic Proof of Theta-Function Identities

Part of the Texts & Monographs in Symbolic Computation book series (TEXTSMONOGR)


This is a tutorial for using two new MAPLE packages, thetaids and ramarobinsids. The thetaids package is designed for proving generalized eta-product identities using the valence formula for modular functions. We show how this package can be used to find theta-function identities as well as prove them. As an application, we show how to find and prove Ramanujan’s 40 identities for his so called Rogers–Ramanujan functions G(q) and H(q). In his thesis Robins found similar identities for higher level generalized eta-products. Our ramarobinsids package is for finding and proving identities for generalizations of Ramanujan’s G(q) and H(q) and Robin’s extensions. These generalizations are associated with certain real Dirichlet characters. We find a total of over 150 identities.

A preliminary version of this paper was presented by J. Frye on January 10, 2013 at JMM2013, San Diego. F. Garvan was supported in part by a grant from the Simon’s Foundation (#318714).

This is a preview of subscription content, log in via an institution.

Buying options

USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
USD   169.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD   219.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions


  1. G.E. Andrews, Ramunujan’s “lost” notebook. III. The Rogers–Ramanujan continued fraction. Adv. Math. 41(2), 186–208 (1981). MR 625893 (83m:10034c)

    Article  MathSciNet  Google Scholar 

  2. A. Berkovich, H. Yesilyurt, On Rogers–Ramanujan functions, binary quadratic forms and eta-quotients. Proc. Am. Math. Soc. 142(3), 777–793 (2014). MR 3148513

    Article  MathSciNet  Google Scholar 

  3. B.C. Berndt, Ramanujan’s Notebooks. Part III (Springer, New York, 1991). MR 1117903

    Book  Google Scholar 

  4. B.C. Berndt, An overview of Ramanujan’s notebooks, Ramanujan: Essays and Surveys. History of Mathematics, vol. 22 (American Mathematical Society, Providence, 2001), pp. 143–164. MR 1862749

    Chapter  Google Scholar 

  5. B.C. Berndt, G. Choi, Y.-S. Choi, H. Hahn, B.P. Yeap, A.J. Yee, H. Yesilyurt, J. Yi, Ramanujan’s forty identities for the Rogers–Ramanujan functions. Mem. Am. Math. Soc. 188(880), vi+96 (2007). MR 2323810

    Article  MathSciNet  Google Scholar 

  6. A.J.F. Biagioli, A proof of some identities of Ramanujan using modular forms. Glasgow Math. J. 31(3), 271–295 (1989). MR 1021804

    Article  MathSciNet  Google Scholar 

  7. B.J. Birch, A look back at Ramanujan’s notebooks. Math. Proc. Camb. Philos. Soc. 78, 73–79 (1975). MR 0379372

    Article  MathSciNet  Google Scholar 

  8. B. Cho, J.K. Koo, Y.K. Park, Arithmetic of the Ramanujan–Göllnitz–Gordon continued fraction. J. Number Theory 129(4), 922–947 (2009). MR 2499414

    Article  MathSciNet  Google Scholar 

  9. F. Garvan, A \(q\)-product tutorial for a \(q\)-series MAPLE package, Sém. Lothar. Combin. 42 (1999). Art. B42d, 27 pp. (electronic), The Andrews Festschrift (Maratea 1998) MR 1701583 (2000f:33001)

    Google Scholar 

  10. H. Göllnitz, Partitionen mit Differenzenbedingungen. J. Reine Angew. Math. 225, 154–190 (1967). MR 0211973

    MathSciNet  MATH  Google Scholar 

  11. B. Gordon, Some continued fractions of the Rogers–Ramanujan type. Duke Math. J. 32, 741–748 (1965). MR 0184001

    Article  MathSciNet  Google Scholar 

  12. S.-S. Huang, On modular relations for the Göllnitz–Gordon functions with applications to partitions. J. Number Theory 68(2), 178–216 (1998). MR 1605895

    Article  MathSciNet  Google Scholar 

  13. T. Huber, D. Schultz, Generalized reciprocal identities. Proc. Am. Math. Soc. 144(11), 4627–4639 (2016). MR 3544515

    Article  MathSciNet  Google Scholar 

  14. D.A. Ireland, A Dirichlet character table generator (2013),

  15. J. Lovejoy, R. Osburn, The Bailey chain and mock theta functions. Adv. Math. 238, 442–458 (2013). MR 3033639

    Article  MathSciNet  Google Scholar 

  16. J. Lovejoy, R. Osburn, Mixed mock modular \(q\)-series. J. Indian Math. Soc. (N.S.) (2013). Special volume to commemorate the 125th birth anniversary of Srinivasa Ramanujan, 45–61. MR 3157335

    Google Scholar 

  17. J. Lovejoy, R. Osburn, \(q\)-hypergeometric double sums as mock theta functions. Pac. J. Math. 264(1), 151–162 (2013). MR 3079764

    Article  MathSciNet  Google Scholar 

  18. J. Lovejoy, R. Osburn, On two 10th-order mock theta identities. Ramanujan J. 36(1–2), 117–121 (2015). MR 3296714

    Article  MathSciNet  Google Scholar 

  19. R.A. Rankin, Modular Forms and Functions (Cambridge University Press, Cambridge, 1977). MR 0498390 (58 #16518)

    Google Scholar 

  20. S. Robins, Arithmetic properties of modular forms, ProQuest LLC, Ann Arbor, MI, 1991. Thesis (Ph.D.)–University of California, Los Angeles. MR 2686433

    Google Scholar 

  21. S. Robins, Generalized Dedekind \(\eta \)-products, The Rademacher Legacy to Mathematics (University Park, PA, 1992). Contemporary Mathematics, vol. 166 (American Mathematical Society, Providence, 1994), pp. 119–128. MR 1284055 (95k:11061)

    Google Scholar 

  22. L. Ye, A symbolic decision procedure for relations arising among Taylor coefficients of classical Jacobi theta functions. J. Symb. Comput. 82, 134–163 (2017). MR 3608235

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to Frank Garvan .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Frye, J., Garvan, F. (2019). Automatic Proof of Theta-Function Identities. In: Blümlein, J., Schneider, C., Paule, P. (eds) Elliptic Integrals, Elliptic Functions and Modular Forms in Quantum Field Theory. Texts & Monographs in Symbolic Computation. Springer, Cham.

Download citation

  • DOI:

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-04479-4

  • Online ISBN: 978-3-030-04480-0

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics