Families Of Multisums As Mock Theta Functions

Early in 1920, Ramanujan defined 17 functions F(q), |q| < 1, which he called mock theta functions. Since then, the study about mock theta functions has attracted a lot of attention of many mathematicians, and they have made great achievements in this area. Many assertions given by Ramanujan have been proved and some other new mock theta functions have also been found. Based on the following three methods: basic hypergeometric series, constant term, and the Bailey lemma, a great deal of deep results about mock theta functions have been obtained. Around the year 2000, the theory of mock theta functions took a new turn through a connection with the Maass form. Now, each mock theta function is to be known as the holomorphic part of a weight 1/2 harmonic weak Maass form with a unary theta function as its shadow. In this thesis, by applying the Bailey chain and relations between Hecke-type double sums and Appell-Lerch sums, we construct families of multisums as mock theta functions.In Chapter 1, we introduce some useful notations and the background of mock theta functions and Bailey pairs. We reveal the research progress about mock theta functions and list our main theorems which are some new families of multisums as mock theta functions.In Chapter 2, by iterating some Bailey pairs along the Bailey chain, we obtain serval q-hypergeometric multisums which can be expressed in terms of Hecke-type double sums and theta functions. Meanwhile, we introduce the recent work given by Lovejoy and Osburn on constructing multisums as mock theta functions by means of the Bailey chain method.In Chapter 3, we give the proofs of our main theorems. Based on the Bailey pairs in Slater's list, we derive some families of multisums as mock theta functions by establishing generalized Bailey pairs with more parameters.In Chapter 4, we give some examples of the main theorems by using some Slater's Bailey pairs. Meanwhile, we study all the other Bailey pairs in Slater's list which can be applied to the main theorems. Furthermore, some identities between new mock theta functions and classical ones are established. Finally,based on an observation about the proofs of the main theorems, many q-series transformations are obtained.