ARITHMETIC PROPERTIES OF PARTITIONS AND HECKE-ROGERS TYPE IDENTITIES

Abstract

This thesis contains four parts on various types of partitions and Hecke-Rogers type identities. 1. We give explicit formulas for the number of partition pairs and triples with 3 cores. From these formulas, we establish many arithmetic identities satisfied by these two partition functions. 2. We prove three infinite families of congruences modulo arbitrary powers of 11 for some partition functions, including 11-regular partitions and 11-core partitions. We also confirm a conjecture of H.H. Chan and P.C. Toh on the ordinary partition function $p(n)$. 3. We introduce a unified modular approach to find $q$-product representations for the generating functions of $k$-colored generalized Frobenius partitions. We give various representations when $k$ is less than 18. Moreover, we discover new surprising properties of the $k$-colored generalized Frobenius partitions. 4. We prove three new Hecke-Rogers type identities. We also provide new proofs to five identities of Ramanujan on false theta functions.