Mock Theta Function Identity And Related Problem

Mock theta functions,as an important branch of basic hypergeometric series,are related to theta series,integer partitions and modular forms.The aim of this dissertation is to study mock theta function identities and related problems,including the single sums and Hecke-type triple sums associated with classical mock theta functions,the relationships as mock theta functions,the extensions of two-parameter mock theta functions.The main results of this dissertation have four aspects as follows.1.The new single sums for classical mock theta functions as well as theta series identities are studied.The main results include that(ⅰ)the single sums for classical mock theta functions are obtained by using Liu’s transformation formula.As a byproduct,the relation connecting a second order mock theta function and four eighth order mock theta functions is proved directly;(ⅱ)in view of the sign flips on these identities,the corresponding theta series identities are deduced by the new Bailey pairs and Watson’s 8?7 transformation formula.Indeed,the truncated forms for theta series identities are given.2.For the triple sums involving mock theta functions,with the help of the Bailey pairs as well as the properties of Hecke-type double sums and triple sums,we present the Hecke-type triple sums for the third order and fifth order mock theta functions.In a similar manner,we arrive at the Hecke-type triple sums for theta series and partition function.3.For the relationships associated with mock theta functions,we establish new identities for second order,sixth order and eighth order mock theta functions by employing the properties of Appell-Lerch sums.On this basis,the known mock theta function identities and theta series identities with sign flips are discussed.4.By Watson’s 8?7 transformation formula and partial fraction decomposition,we provide several new families of identities involving two-parameter mock theta functions and express them in terms of Appell-Lerch sums.Specializations of parameters may deduce a series of new two-parameter mock theta functions.Moreover,we conclude the plenty of identities involving classical mock theta functions as well as theta series.