Evaluation of Hecke-Rogers series and expansions of the rank function (Q6645981)

The paper evaluates the double sum \[\sum_{n=-\infty}^\infty \sum_{j=0}^{kn} q^{\frac{an(n-1 )}{2}-\frac{bj(j-1)}{2}}x^jy^n\] for \(a,b,k\) positive integers with \(a>bk^2\) in terms of theta functions and Appell-Lerch functions. A special case evaluates the sum \[\sum_{n=1}^\infty\sum_{m=1}^n q^{ 2n^2-\frac{m(3m-1) }{2}} (1-q^{2m-1})(x^n+x^{-n}).\] This evaluation includes several known identities for the mock theta function.