1. Introduction

In his last letter to G.H. Hardy[11], S. Ramanujan listed seventeen mock theta functions of order three, five, five and seven. According to Ramanujan a mock theta function is a function satisfying the following two conditions:

(0) For every root of unity , there is a -function such that the difference is bounded as radially.

(1) There is no single -function which works for all i.e., for every -function there is some root of unity for which is unbounded as radially.

G.N. Watson[17] found three more mock theta functions of order three. In his “Lost” Notebook Ramanujan gave six more mock theta functions which were called by G.E. Andrews and D. Hickerson[5] of order six and four mock theta functions which were called by Choi[6] of order ten. B. Gordon and R.J. McIntosh[8] generated eight mock theta functions and called them of order eight, but four of them were later found of lower order. Hikami[9] found one more mock theta function of order two.

Recently Andrews[2] in his paper generated some new mock theta functions and found four of them interesting. Bringmann, Hikami and Lovejoy developed two more mock theta functions. We in[12-14] have made a comprehensive study of these mock theta functions.

After studying these mock theta functions in[12-14] I started considering their bilateral form. The study became interesting as in their bilateral form the mock theta functions

The four mock theta functions of Andrews [2]:

(1.1)

(1.2)

(1.3)

and

(1.4)

Andrews called as a companion to the third order mock theta function of Ramanujan

The two mock theta functions of Bringmann, Hikami and Lovejoy [10]:

(1.5)

and

(1.6)

We now write the mock theta function as basic bilateral series, and following Watson, call them ’Complete’.

(1.7)

(1.8)

(1.9)

and

(1.10)

The scheme of the paper is as follows:

In section 3, we give an expansion for these functions using Slater’s expansion formula[7] and show their relationship with other bilateral mock theta functions.

In section 4 and section 5, we express these ‘complete’ mock theta functions as series.

In section 6, we represent these ‘complete’ mock theta functions as continued fractions.

In section 7, we give alternative definitions, using Bailey’s bilateral transformation[7].

In section 8, relationship between two sets of bilateral mock theta functions is shown.

In section 9 and 10, we define extended form for the ‘complete’ mock theta functions and show that they are - functions and certain relationship between these functions.

2. Basic Standard Results

We shall use the following usual basic hypergeometric notations:

q^k < 1, n, non negative integer

,

A generalized basic bilateral series is defined as

where

3. Relations between Bilateral Mock Theta Functions

Taking r = 2 in the following expansion of Slater [7, (5.4.3), p. 129]:

(3.1)

where , , and idem after the expression means that the preceding expression is repeated with and interchanged.

We now give expansions for these bilateral mock theta functions.

(i) Letting ,and taking in (3.1), we have

(3.2)

(3.3)

, in (3.1))

(3.4)

(3.5)

Relations between bilateral mock theta functions Special Cases

Take in (3.2) :

(3.6)

Take in (3.3):

(3.7)

Take in (3.4):

(3.8)

Take in (3.5):

(3.9)

4. Expansion of Bilateral Mock Theta Functions and as Series

In (3.1) we take to get [7, (5.4.5), p. 130]

(4.1)

where .

We will now specialize the paramaeter in (4.1) to get expansions of bilateral mock theta functions as q- hypergeometric series.

(i) Letting and in (4.1), to get

(4.2)

(4.3)

and in (4.1))

5. Expansions of Bilateral Mock Theta Functions and as Series

For getting the expansion of a bilateral series as a series, we shall use the following expansion formula, which is obtained by putting in Slater’s expansion formula (3.1) [7, (5.4.4), p. 130]:

(5.1)

(i) Letting and in (5.1), we have

(5.2)

(5.3)

( and in (5.1))

6. Bilateral or ‘Complete’ Mock Theta Functions as Continued Fractions

We first write the bilateral as the sum of two basic hypergeometric series and then use the continued fraction [1,(3.79) p 82]:

(6.1)

to represent as a continued fraction.

(i) Now (4.2) gives

(6.2)

where

and

Dividing the expressions on both sides of (6.2) by the first summation on the right side of (6.2), we have

(6.3)

Letting in (6.1), and then putting it in the quotient of the summation on the right side of (6.3) to have

(ii) Similarly by (4.3), we have

where

and

.

Hence we have

(6.5)

(iii) For we will have the same continued fraction by (8.2).

(iv) For we will have the same continued fraction by (8.4).

7. Alternative Definitions for ‘Complete’ Mock Theta Functions

We shall use the following bilateral transformation of Bailey [7, 5.20(ii), p 137] to get alternative definition :

(7.1)

and

(7.2)

(i) Let and take , in (7.1) and (7.2), respectively, to get

(7.3)

and

(7.4)

(7.5)

in (7.1))

(7.6)

in 7.2))

(7.7)

in (7.1))

(7.8)

in (7.2))

(7.9)

in (7.1))

(7.10)

in (7.2))

8. Relationship between Bilateral Mock Theta Functions

From these alternative definition it is interesting to note that bilateral mock theta functions developed by Andrews can be expressed as bilateral mock theta functions developed by Bringmann et al.

From (7.3)

(8.1)

From (7.4)

(8.2)

From (7.5)

(8.3)

From (7.6)

(8.4)

From (7.7)

(8.5)

From (7.8)

(8.6)

From (7.9)

(8.7)

From (7.10)

(8.8)

and are eighth order mock theta functions given by Gordon and McIntosh [8].

9. Extended Form for Bilateral ‘Complete’ Mock Theta Functions

We define the extended ‘Complete’ mock theta functions:

(9.1)

(9.2)

(9.3)

and

(9.4)

For these extended functions reduce to the bilateral mock theta functions.

We show they are -functions. Before we prove they are -functions, we define a-function.

Truesdell[15] calls the functions which satisfy the functional equation

,

F-functions.

The q-analogue is that the functions which satisfy the functional equation

,where

are called -Functions.

By definition

Hence

Thus is a -function.

Similarly, all other functions listed in section 9 are - functions.

10. Relations between Extended Functions

Now we give a relationship between these generalized mock theta functions.

11. Conclusions

The study of basic bilateral mock theta functions is interesting as it gives alternative forms and close relationship among the mock theta functions. We have expressed them as series and represented them as continued fractions. The same method can be applied for other classical mock theta functions

I have considered these mock theta functions (1.1)--(1.6) in detail finding their properties and expansions etc [12], [13] and [14].