Proofs of some conjectures of Chan on Appell-Lerch sums

On page 3 of his lost notebook, Ramanujan defines the Appell-Lerch sum phi(q) := Sigma(infinity)(n=0) (-q; q)(2n)q(n+1)/(q; q(2))(n+1)(2) which is connected to some of his sixth order mock theta functions. Let Sigma(infinity)(n=1) 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) equivalent to a(50n + 39) equivalent to a(50n + 49) equivalent to 0 (mod 25) as well as a(1250n + 250r + 219) equivalent to 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.