Proofs of some conjectures of Chan on Appell–Lerch sums

  • Nayandeep Deka BaruahEmail author
  • Nilufar Mana Begum


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.


Appell–Lerch sum Theta function Mock theta function Congruence 

Mathematics Subject Classification

Primary 11P83 Secondary 33D15 



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Authors and Affiliations

  1. 1.Department of Mathematical SciencesTezpur UniversitySonitpurIndia

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