An Extension of the Hardy-ramanujan Circle Method and Applications to Partitions without Sequences

We develop a generalized version of the Hardy-Ramanujan “circle method” in order to derive asymptotic series expansions for the products of modular forms and mock theta functions. Classical asymptotic methods (including the circle method) do not work in this situation because such products are not modular, and in fact, the “error integrals” that occur in the transformations of the mock theta functions can (and often do) make a significant contribution to the asymptotic series. The resulting series include principal part integrals of Bessel functions, whereby the main asymptotic term can also be identified. To illustrate the application of our method, we calculate the asymptotic series expansion for the number of partitions without sequences. Andrews showed that the generating function for such partitions is the product of the third order mock theta function χ and a (modular) infinite product series. The resulting asymptotic expansion for this example is particularly interesting because the error integrals in the modular transformation of the mock theta function component appear in the exponential main term.