Skip to main content
SpringerLink
Log in

Proofs of some conjectures of Chan on Appell–Lerch sums

  • Published:
The Ramanujan Journal Aims and scope Submit manuscript

Abstract

On page 3 of his lost notebook, Ramanujan defines the Appell–Lerch sum

$$\begin{aligned} \phi (q):=\sum _{n=0}^\infty \dfrac{(-q;q)_{2n}q^{n+1}}{(q;q^2)_{n+1}^2}, \end{aligned}$$

which is connected to some of his sixth order mock theta functions. Let \(\sum _{n=1}^\infty a(n)q^n:=\phi (q)\). In this paper, we find a representation of the generating function of \(a(10n+9)\) in terms of q-products. As corollaries, we deduce the congruences \(a(50n+19)\equiv a(50n+39)\equiv a(50n+49)\equiv 0~(\text {mod}~25)\) as well as \(a(1250n+250r+219)\equiv 0~(\text {mod}~125)\), where \(r=1\), 3, and 4. The first three congruences were conjectured by Chan in 2012, whereas the congruences modulo 125 are new. We also prove two more conjectural congruences of Chan for the coefficients of two Appell–Lerch sums.

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. Andrews, G.E., Hickerson, D.: Ramanujan’s ‘lost’ notebook VII: the sixth order mock theta functions. Adv. Math. 89, 60–105 (1991)

    Article  MathSciNet  Google Scholar 

  2. Appell, P.: Sur les fonctions doublement périodiques de troisième espèce. Ann. Sci. Éc. Norm. Supér. 1, 135–164 (1884)

    Article  Google Scholar 

  3. Appell, P.: Développements en série des fonctions doublement périodiques de troisième espèce. Ann. Sci. Éc. Norm. Supér. 2, 9–36 (1885)

    Article  MathSciNet  Google Scholar 

  4. Appell, P.: Sur les fonctions doublement périodiques de troisi‘eme esp‘ece. Ann. Sci. Éc. Norm. Supér. 3, 9–42 (1886)

    Article  Google Scholar 

  5. Baruah, N.D., Begum, N.M.: On exact generating functions for the number of partitions into distinct parts. Int. J. Number Theory 14, 1995–2011 (2018)

    Article  MathSciNet  Google Scholar 

  6. Berndt, B.C.: Number Theory in the Spirit of Ramanujan. American Mathematical Society, Providence, RI (2006)

    Book  Google Scholar 

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

    Book  Google Scholar 

  8. Chan, S.H.: Congruences for Ramanujan’s $\phi $ function. Acta Arith. 153, 161–189 (2012)

    Article  MathSciNet  Google Scholar 

  9. Chan, S.H., Mao, R.: Two congruences for Appell-Lerch sums. Int. J. Number Theory 8, 111–123 (2012)

    Article  MathSciNet  Google Scholar 

  10. Choi, Y.-S.: Identities for Ramanujan’s sixth-order mock theta functions. Q. J. Math. 53, 147–159 (2002)

    Article  MathSciNet  Google Scholar 

  11. Ding, H., Xia, E.X.W.: Arithmetric properties for Appell-Lerch sums (Preprint)

  12. Hickerson, D.R., Mortenson, E.T.: Hecke-type double sums, Appell-Lerch sums, and mock theta functions, I. Proc. Lond. Math. Soc. 109, 382–422 (2014)

    Article  MathSciNet  Google Scholar 

  13. Hickerson, D.R., Mortenson, E.T.: Dyson’s ranks and Appell-Lerch sums. Math. Ann. 367, 373–395 (2017)

    Article  MathSciNet  Google Scholar 

  14. Hikami, K.: Transformation formula of the “second” order mock theta function. Lett. Math. Phys. 75, 93–98 (2006)

    Article  MathSciNet  Google Scholar 

  15. Lerch, M.: Poznámky k theorii funkcĺ elliptických. Prag. České Ak. Fr. Jos. Rozpr. 24, 465–480 (1892)

    Google Scholar 

  16. Mortenson, E.T.: Ramanujan’s $_{1}\psi _{1}$ summation, Hecke-type double sums, and Appell-Lerch sums. Ramanujan J. 29, 121–133 (2012)

    Article  MathSciNet  Google Scholar 

  17. Mortenson, E.T.: On the dual nature of partial theta functions and Appell-Lerch sums. Adv. Math. 264, 236–260 (2014)

    Article  MathSciNet  Google Scholar 

  18. Ramanujan, S.: Some properties of $p(n)$, the number of partitions of $n$. Proc. Camb. Philos. Soc. 19, 207–210 (1919)

    MATH  Google Scholar 

  19. Ramanujan, S.: The Lost Notebook and Other Unpublished Papers. Narosa, New Delhi (1988)

    MATH  Google Scholar 

  20. Qu, Y.K., Wang, Y.J., Yao, O.X.M.: Generalizations of some conjectures of Chan on congruences for Appell-Lerch sums. J. Math. Anal. Appl. 460, 232–238 (2018)

    Article  MathSciNet  Google Scholar 

  21. Waldherr, M.: On certain explicit congruences for mock theta functions. Proc. Am. Math. Soc. 139, 865–879 (2011)

    Article  MathSciNet  Google Scholar 

  22. Zwegers, S.P.: Mock theta functions. Ph. D. Thesis, Universiteit Utrecht (2002)

Download references

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, Tezpur University, Sonitpur, Assam, 784028, India

    Nayandeep Deka Baruah & Nilufar Mana Begum

Authors
  1. Nayandeep Deka Baruah
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Nilufar Mana Begum
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Nayandeep Deka Baruah.

Additional information

Publisher's Note

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

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Baruah, N.D., Begum, N.M. Proofs of some conjectures of Chan on Appell–Lerch sums. Ramanujan J 51, 99–115 (2020). https://doi.org/10.1007/s11139-018-0076-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11139-018-0076-x

Keywords

Mathematics Subject Classification

Search

Navigation