Dissections of Strange q-Series

In a study of congruences for the Fishburn numbers, Andrews and Sellers observed empirically that certain polynomials appearing in the dissections of the partial sums of the Kontsevich–Zagier series are divisible by a certain q-factorial. This was proved by the first two authors. In this paper, we extend this strong divisibility property to two generic families of q-hypergeometric series which, like the Kontsevich–Zagier series, agree asymptotically with partial theta functions.