Proofs of some conjectures of Chan on Appell–Lerch sums

• Published in: The Ramanujan journal Vol. 51; no. 1; pp. 99 - 115
• Main Authors: Baruah, Nayandeep Deka, Begum, Nilufar Mana
• Format: Journal Article
• Language: English
• Published: New York Springer US 2020; Springer Nature B.V
• Subjects: Combinatorics; Congruences; Field Theory and Polynomials; Fourier Analysis; Functions of a Complex Variable; Mathematics; Mathematics and Statistics; Number Theory; Sums; Mock theta function; Theta function; Congruence; Primary 11P83; Appell–Lerch sum; Secondary 33D15
• ISSN: 1382-4090; 1572-9303
• DOI: 10.1007/s11139-018-0076-x

On page 3 of his lost notebook, Ramanujan defines the Appell–Lerch sum ϕ ( q ) : = ∑ n = 0 ∞ ( - q ; q ) 2 n q n + 1 ( q ; q 2 ) n + 1 2 , which is connected to some of his sixth order mock theta functions. Let ∑ n = 1 ∞ a ( n ) q n : = ϕ ( q ) . In this paper, we find a representation of the generating function of a ( 10 n + 9 ) in terms of q -products. As corollaries, we deduce the congruences a ( 50 n + 19 ) ≡ a ( 50 n + 39 ) ≡ a ( 50 n + 49 ) ≡ 0 ( mod 25 ) as well as a ( 1250 n + 250 r + 219 ) ≡ 0 ( 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.