Relations Between the Rank and the Crank Modulo 9

In [10] various conjectural identities between the ranks and the cranks of partitions modulo 8, 9 and 12 were proposed. These conjectures have now all been proved [14, 15, 11], with three exceptions. Here, we prove these three exceptions. Methods similar to those used here may be used to establish all the theorems and conjectures of [10].