Ramanujan's ${}_1{\psi }_1$ summation, Hecke-type double sums, and Appell-Lerch sums

We use a specialization of Ramanujan’s ${}_1{\psi }_1$ summation to give a new proof of a recent formula of Hickerson and Mortenson which expands a special family of Hecke-type double sums in terms of Appell-Lerch sums and theta functions.