Contributions to the Theory of Q-Series and Mock Theta Functions

Abstract

In Chapter 2, we prove, for the first time, a series of four related identities from Ramanujan's lost notebook. These are one-parameter identities which would directly imply some of the relations for the third order mock theta functions given by Ramanujan. In Chapter 3, we prove very general theorems on the periodicity of signs of the Taylor series coefficients of a large class of q-products. S. Ramanujan, B. Richmond, G. Szekeres, G. E. Andrews, K. G. Ramanathan, M. D. Hirschhorn, and others have examined these types of q-products with the goals of either producing asymptotic expansions or examining the periodicity of signs for their Taylor series coefficients. Further motivation arises from the fact that two of these products yield representations for the famous Rogers-Ramanujan continued fraction and the Ramanujan-Gollnitz-Gordon continued fraction. Our theorems greatly generalize the theorems of Andrews, Hirschhorn, and Ramanathan, and also have an application to another continued fraction of Ramanujan. An interesting representation for q;q10infinity is given in Chapter 4. This representation easily leads to a short proof of Ramanujan's famous congruence p(11n + 6) ≡ 0 (mod 11), where p(n) denotes the number of unrestricted partitions of the positive integer n. In the last chapter, we give simple elementary proofs, with methods known to Ramanujan, of several of Ramanujan's 40 identities for the Rogers-Ramanujan functions. These identities have been previously proved by either L. J. Rogers, G. N. Watson, D. Bressoud, or A. J. F. Biagioli. However, it has not been realized by these authors that several of the identities are related. We also demonstrate how these ideas can be utilized to obtain new identities for the Rogers-Ramanujan functions.