Proofs of Some Conjectures of Chan on Appell-Lerch Sums

Nayandeep Deka Baruah, Nilufar Mana Begum

On page 3 of his lost notebook, Ramanujan defines the Appell-Lerch sum $$\varphi (q):=\sum _{n=0}^{\infty }\frac{(-q;q{)}_{2n}{q}^{n+1}}{(q;q^2{)}_{n+1}^2},$$ which is connected to some of his si…