On ranks of partitions and congruences of special functions

Abstract

This thesis focuses on the rank statistic of partition functions, congruences and relating identities of special functions such as Appell-Lerch sums and partition pairs. Most results in Chapter 2, 3, 4, 5 are reproduced from [58], [59], [25], [24], respectively. F. J. Dyson conjectured that the rank of partitions provides combinatorial interpretations of S. Ramanujan’s famous congruences for partition functions modulo 5 and 7. This together with some other identities between ranks of partitions modulo 5 and 7 were proved by A. O. L. Atkin and H. P. F. Swinnerton-Dyer. In Chapter 2, we prove identities for ranks of partition modulo 10. With a similar method, we obtain identities between the M2-rank of partitions without repeated odd parts modulo 6 and 10 in Chapter 3. A series of identities and congruences of Appell-Lerch sums were discovered by S. H. Chan recently. In Chapter 4, we give a generalization of Chan’s results and also find a new series of identities for Appell-Lerch sums. As special cases, we prove congruences for some mock theta functions. In Chapter 5, we prove two identities related to overpartition pairs. One of them gives a generalization of an identity due to J. Lovejoy, which was used in a joint work by K. Bringmann and Lovejoy to derive congruences for overpartition pairs. We apply our two identities of pairs of partitions where each partition has no repeated odd parts. We also present three partition statistics that give combinatorial explanations of a congruence modulo 3 satisfied by these partition pairs.