Research On The Third-order Mock Theta Function

The work of Ramanujan has had a profound impact on many areas of mathematics,among which research on mock theta functions is very active at present.In this thesis,we mainly study some problems related to the third-order mock theta functions.This thesis consists of five chapters.In Chapter 1,we first give some preliminary knowledge which will be used.Then,a detailed introduction to the background of mock theta functions is presented.At last,we describe in detail the research status of bilateral series of mock theta functions and their duality.In Chapter 2,we mainly derive the different types of bilateral series for the third-order mock theta functions.As applications,some identities between the two-group bilateral series are obtained.Then,we consider duals of the second type in terms of Appell-Lerch sums and duals in terms of partial theta functions defined by Hickerson and Mortenson of duals of the second type in terms of Appell-Lerch sums of such bilateral series associated to some third-order mock theta functions.In Chapter 3,motivated by the work of Andrews in[4],we study a transformation of bilateral series again.Using this transformation,the generalizations of theorems involving some third-order mock theta functions are derived.In Chapter 4,we first give some representations for four new mock theta functions defined by Andrews[11]and Bringmann,Hikami and Lovejoy[22]using divisor sums.Then,some transformation and summation formulae for these functions and corresponding bilateral series are derived by using_2?2 series ??? and Ramanujan's sum ???In Chapter 5,in terms of several transformation formulae for basic hypergeometric series,we find some new series representations for four third order mock theta functions and give the extensions of Mortenson's identities.Meanwhile,we offer a simple proof of Ramanujan's_1?1 summation formula.