Skip to main content
SpringerLink
Log in

On 5- and 10-dissections for some infinite products

  • Published:
The Ramanujan Journal Aims and scope Submit manuscript

Abstract

Quite recently, Xia and Zhao established the 10-dissections for Hirschhorn’s two infinite q-series products by using two MAPLE packages and the theory of modular forms. Utilizing the Jacobi triple product identity, we not only establish the 10-dissections for two infinite q-series products, introduced by Baruah and Kaur, but give an elementary proof of the 10-dissections due to Xia and Zhao. Moreover, we obtain the 5-dissections for four quotients of infinite q-series products related to the Rogers–Ramanujan functions. Using these dissections, the coefficients in these series expansions have periodic sign patterns with a few exceptions.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Alladi, K., Gordon, B.: Vanishing coefficients in the expansion of products of Rogers–Ramanujan type. In: Andrews, G.E., Bressoud, D. (eds.) Proceedings of the Rademacher Centenary Conference, Contemporary Mathematics, vol. 166, pp. 129–139 (1994)

  2. Andrews, G.E.: Ramanujan’s “lost” notebook III. The Rogers–Ramanujan continued fraction. Adv. Math. 41, 186–208 (1981)

    Article  Google Scholar 

  3. Andrews, G.E., Bressoud, D.M.: Vanishing coefficients in infinite product expansions. J. Aust. Math. Soc. Ser. A 27(2), 199–202 (1979)

    Article  MathSciNet  Google Scholar 

  4. Baruah, N. D., Kaur, M.: Some results on vanishing coefficients in infinite product expansions. Ramanujan J. (2019) (in press). https://doi.org/10.1007/s11139-019-00172-x

  5. Berndt, B.C.: Ramanujan’s Notebooks, Part III. Springer, New York (1991)

    Book  Google Scholar 

  6. Chan, S.H., Yesilyurt, H.: The periodicity of the signs of the coefficients of certain infinite products. Pac. J. Math. 225(1), 13–32 (2006)

    Article  MathSciNet  Google Scholar 

  7. Chen, S.-D., Huang, S.-S.: On the series expansion of the Göllnitz-Gordon continued fraction. Int. J. Number Theory 1(1), 53–63 (2005)

    Article  MathSciNet  Google Scholar 

  8. Chern, S.: Asymptotics for the Taylor coefficients of certain infinite products. Ramanujan J. (2020). https://doi.org/10.1007/s11139-020-00273-y

  9. Chern, S., Tang, D.: Vanishing coefficients in quotients of theta functions of modulus five. Bull. Aust. Math. Soc. (2020) (in press). https://doi.org/10.1017/S0004972720000271

  10. Chern, S., Tang, D.: 5-Dissections and sign patterns of Ramanujan’s parameter and its companion. To appear in Czechoslovak Math. J

  11. Cooper, S.: Ramanujan’s Theta Functions. Springer, Cham (2017)

    Book  Google Scholar 

  12. Dou, D.Q.J., Xiao, J.: The 5-dissections of two infinite product expansions. Ramanujan J. (2020) (in press). https://doi.org/10.1007/s11139-019-00200-w

  13. Frye, J., Garvan, F.: Automatic proof of theta-function identities. In: Elliptic Integrals, Elliptic Functions and Modular Forms in Quantum Field Theory, pp. 195–258. Springer, Cham (2019)

  14. Hirschhorn, M.D.: On the expansion of Ramanujan’s continued fraction. Ramanujan J. 2(4), 521–527 (1998)

    Article  MathSciNet  Google Scholar 

  15. Hirschhorn, M.D.: On the expansion of a continued fraction of Gordon. Ramanujan J. 5(4), 369–375 (2011)

    Article  MathSciNet  Google Scholar 

  16. Hirschhorn, M.D.: The Power of \(q\), Developments in Mathematics, vol. 49. Springer, Cham (2017)

    Google Scholar 

  17. Hirschhorn, M.D.: Two remarkable \(q\)-series expansions. Ramanujan J. 49(2), 451–463 (2018)

    Article  MathSciNet  Google Scholar 

  18. Hirschhorn, M.D., Roselin, M.D.: On the 2-, 3-, 4- and 6-Dissections of Ramanujan’s Cubic Continued Fraction and Its Reciprocal, Ramanujan Rediscovered: Proceedings of a Conference on Elliptic Functions, Partitions, and \(q\)-Series, in Memory of K. Venkatachaliengar, Ramanujan Mathematical Society, Bangalore, June 1–5, 2009, India, pp. 125–138 (2010)

  19. Kadell, K.W.J.: An injection for the Ehrenpreis Rogers-Ramanujan problem. J. Comb. Theory Ser. A 86(2), 390–394 (1999)

    Article  MathSciNet  Google Scholar 

  20. Lin, B.L.S.: On the expansion of a continued fraction of order twelve. Int. J. Number Theory 9(8), 2019–2031 (2013)

    Article  MathSciNet  Google Scholar 

  21. Mc Laughlin, J.: Further results on vanishing coefficients in infinite product expansions. J. Austral. Math. Soc. Ser. A 98(1), 69–77 (2015)

    Article  MathSciNet  Google Scholar 

  22. Ramanathan, K.G.: Generalisations of some theorems of Ramanujan. J. Number Theory 29(2), 118–137 (1988)

    Article  MathSciNet  Google Scholar 

  23. Richmond, B., Szekeres, G.: The Taylor coefficients of certain infinite products. Acta Sci. Math. 40(3–4), 347–369 (1978)

    MathSciNet  MATH  Google Scholar 

  24. Tang, D.: Vanishing coefficients in some \(q\)-series expansions. Int. J. Number Theory 15(4), 763–773 (2019)

    Article  MathSciNet  Google Scholar 

  25. Tang, D.: Vanishing coefficiets in four quotients of infinite product expansions. Bull. Aust. Math. Soc. 100(2), 216–224 (2019)

    Article  MathSciNet  Google Scholar 

  26. Tang, D., Xia, E.X.W.: Several \(q\)-series related to Ramnujan’s theta functions. Ramanujan J. (2019) (in press). https://doi.org/10.1007/s11139-019-00187-4

  27. Vanitha, A.: On the expansion of Ramanujan’s continued fraction of order sixteen. Tamsui Oxf. J. Inf. Math. Sci. 30(1), 81–99 (2016)

    MathSciNet  Google Scholar 

  28. Xia, E.X.W., Yao, O.X.M.: On 2- and 4-dissections for some infinite products. Rocky Mt. J. Math. 43(6), 2033–2047 (2013)

    Article  MathSciNet  Google Scholar 

  29. Xia, E.X.W., Yao, O.X.M.: 4-Dissections and 8-dissections for some infinite products. Rocky Mt. J. Math. 44(5), 1685–1696 (2014)

    Article  MathSciNet  Google Scholar 

  30. Xia, E.X.W., Zhao, A.X.H.: Generalizations of Hirschhorn’s results on two remarkable \(q\)-series expansions. Exp. Math. (2020) (in press). https://doi.org/10.1080/10586458.2020.1712565

Download references

Acknowledgements

The author would like to acknowledge Prof. Shaoshi Chen at Chinese Academy of Sciences for hosting him for several days in November 2019, during which parts of the present research were performed. The author would also like to thank Prof. Mike Hirschhorn and Prof. Shishuo Fu for their sincere help and sustaining encouragement. The author would also like to acknowledge the referee for his/her careful reading and helpful comments.

Author information

Authors and Affiliations

  1. Center for Applied Mathematics, Tianjin University, Tianjin, 300072, People’s Republic of China

    Dazhao Tang

Authors
  1. Dazhao Tang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dazhao Tang.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This work was supported by the Postdoctoral Science Foundation of China (No. 2019M661005).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Tang, D. On 5- and 10-dissections for some infinite products. Ramanujan J 56, 425–450 (2021). https://doi.org/10.1007/s11139-020-00340-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11139-020-00340-4

Keywords

Mathematics Subject Classification

Search

Navigation