Representations Of Mock Theta Functions

Motivated by the works of Liu,we provide a unified approach to find Appell-Lerch series and Hecke-type series representations for mock theta functions.We es-tablish a number of parameterized identities with two parameters a and b.Specializing the choices of?a,b?,we not only give various known and new representations for the mock theta functions of orders 2,3,5,6 and 8,but also present many other interesting identities.We find that some mock theta functions of different orders are related to each other,in the sense that their representations can be deduced from the same?a,b?-parameterized identity.Furthermore,we introduce the concept of false Appell-Lerch series.We then express the Appell-Lerch series,false Appell-Lerch series and Hecke-type series in this paper using the building blocks m?x,q,z?and fa,b,c?x,y,q?introduced by Hickerson and Mortenson,as well as ????x,q,z?and ???_a,b,c?x,y,q?introduced in this paper.We also show the equivalences of our new representations for several mock theta functions and the known representations.The paper is organized as follows.In Section 1.2,we first recall some formulas from the theory of basic hypergeometric series.We also discuss some limiting cases of Watson's q-analog of Whipple's theorem.The formulas listed in Section 1.2.1 will be used in evaluating certain terminated ₃?₂ series,which are fundamental for establish-ing parameterized identities.Then in Section 1.2.2 we give the definitions and useful properties of the building blocks of Appell-Lerch series,false Appell-Lerch series and Hecke-type series.As examples,we will rewrite the identities?1.1.6?,?1.1.8?and?1.1.9?using these building blocks.In Section 1.2.3,we shall first prove Theorems 1.1and 1.2.Then as first examples of these theorems,we give several parameterized iden-tities and new Hecke-type identities.We continue to apply Theorem 1.1 in Sections2-6,where we discuss mock theta functions of orders 2,3,5,6 and 8,respectively.For each mock theta function,we correspondingly establish some parameterized iden-tities.Each of these identities gives us a representation for the mock theta function and produces new interesting identities.