| This thesis explores three related topics in number theory, cranks of partitions, generalized Lambert series, and basic hypergeometric series, and they are discussed in Chapters 2, 3, and, 4 and 5, respectively.; The existence of the crank, a statistic for the partition function p(n), was first conjectured by F. J. Dyson in 1944 and was later established by G. E. Andrews and F. G. Garvan in 1987. However, much earlier, in his lost notebook, Ramanujan studied the generating function F a(q) for the crank and offered several elegant claims about it, although it seems unlikely that he was familiar with all the combinatorial implications of the crank. In particular, Ramanujan found several congruences for Fa(q) in the ring of formal power series in the two variables a and q. In Chapter 2, using an obscure identity found on page 59 of the lost notebook, we provide uniform proofs of several congruences in the ring of formal power series for the generating function Fa(q) of cranks. These congruences are found, sometimes in abbreviated form, in the lost notebook, and imply dissections of Fa(q). Consequences of our work are interesting new q-series identities and congruences in the spirit of Atkin and Swinnerton-Dyer.; In Chapter 3, we present two infinite families of generalized Lambert series identities, and deduce several known identities from them. They include an identity due to M. Jackson, a corollary of Ramanujan's 1psi 1 summation formula, and a recent identity of G. E. Andrews, R. P. Lewis, and Z.-G. Liu.; In Chapters 4 and 5, we apply the same method used in Chapter 3 on basic hypergeometric series. In Chapter 4, we present a new proof of Ramanujan's 10, summation formula. And finally, in Chapter 5, we present a new proof of a general transformation formula for basic hypergeometric series that was discovered by D. B. Sears in 1951. We also discuss some special cases. |
