#### Abstract

We consider the second-order mock theta function (), which Hikami came across in his work on mathematical physics and quantum invariant of three manifold. We give their bilateral form, and show that it is the same as bilateral third-order mock theta function of Ramanujan. We also show that the mock theta function () outside the unit circle is a theta function and also write as a coefficient of of a theta series. First writing as a coefficient of a theta function, we prove an identity for .

#### 1. Brief History of Mock Theta Functions

The mock theta functions were introduced and named by Ramanujan and were the subjects of Ramanujanβs last letter to Hardy, dated January 12, 1920, to be specific [1, 2]. Ramanujan gave a list of seventeen functions which he called βmock theta functions.β He divided them into four groups of functions of order 3, 5, 5, and 7. Ramanujan did not rigorously define a mock theta function nor he define the order of a mock theta function. A definition of the order of a mock theta function is given in the Gordon-McIntosh paper on modular transformation of Ramanujanβs fifth and seventh-order mock theta functions [3] Watson [4] while constructing transformation laws for the mock theta function found three further mock theta functions of order 3.

In 1976, Andrews while visiting Trinity college, Cambridge, discovered in the mathematical library of the college a notebook written by Ramanujan towards the end of his life and Andrews called it βLostβ Notebook. In the lost notebook were six more mock theta functions and linear relation between them. Andrews and Hickerson [5] called these mock theta functions of sixth-order and proved the identities.

In the βLostβ Notebook on page 9 appear four more mock theta functions which were called by Choi of tenth-order. Ramanujan also gave eight linear relations connecting these mock theta functions of tenth-order and these relations were proved by Choi [6].

Gordon and McIntosh listed eight functions in their eighth-order paper [7], but later, in their survey paper [8], classified only four of them as eighth-order. The other four are more simple in their modular transformation laws and therefore are considered to be of lower order.

We now come to the second-order mock theta functions. McIntosh [9] considered three second-order mock theta functions and gave transformation formulas for them. Hikami [10] in his work on mathematical physics and quantum invariant of three manifold came across the *q*-series:
and proved that is a mock theta function and called it of β2ndβ order.

He further showed that is a sum of two mock theta functions and where is of second-order and is Ramanujanβs mock theta function of third-order. This will be the basis of our study in this paper.

Before we begin with the study of and it will be appropriate to mention the work done earlier.

Gordon and McIntosh in their survey paper [8] have shown that is essentially the odd part of the second-order mock theta function _{, }which appears as in Andrewsβ paper on Mordell integrals and Ramanujanβs lost notebook [11] and also in McIntosh paper on second-order mock theta functions [9]. In particular,

where

Since the even part of is the ordinary theta function

it follows that the odd part and are second-order mock theta functions. Thus is a linear combination of second-order and third-order mock theta function. In some sense, mock theta functions of orders 1, 2, 3, 4, and 6 are all in the same family.

The paper is divided as follows.

In Section 3 we expand as a bilateral *q*-series and show that it is also a sum of the second-order mock theta function and the third-order mock theta function . By using Baileyβs transformation we have the interesting result that the bilateral is the same as the bilateral _{.}

In Section 4, using bilateral transformation of Slater, we write as a bilateral series series with a free parameter *c*.

In Section 5, a mild generalization of is given and we show that this generalized function is a -function.

In Section 6 we show that , outside the unit circle , is a theta function.

In Section 7 we state a generalized Lambert Series expansion for as given in [8].

In Section 8 we show that is a coefficient of of a theta function.

In Section 9 we prove an identity for using as a coefficient of of a theta function.

In Section 10 a double series expansion for is obtained by using Bailey pair method.

#### 2. Basic Preliminaries

We first introduce some standard notation.

If and are complex numbers with and is a nonnegative integer, then

Ramanujanβs mock theta function of third-order and is

We will use the following notations for -functions.

*Definition 2.1. *If and then
If is a positive integer and is an integer,
By Jacobiβs triple product identity [12, page 282]

##### 2.1. More Definitions

If is a complex number with , then

If *s* is an integer, then

Using these definitions,

We shall use the following theorems.

Theorem 2.2 (see [13, Theorem , page 644]). *Let be fixed, . Let , and be fixed integers with and . Define
**
Then is meromorphic for , with simple poles at all points such that for some integer . The residue of at such a point is
*

Theorem 2.3 (see [13, Theorem (a), page 647]). *Suppose that
**
for all and that satisfies
**
where and . Then
**
Truesdell [14] calls the functions which satisfy the difference equation
**
as -function. He unified the study of these -functions. **The functions which satisfy the -analogue of the difference equation **
where
**
are called -functions.*

#### 3. Bilateral as a Sum of Two Mock Theta Functions of Different Orders

(i) We shall denote the bilateral of by . We define it as

Now

and we use (1.2) in the first summation and (2.2) in the second summation, to write

Thus is a sum of a second-order mock theta function and a third-order mock theta function.

(ii) Transformation of Bilateral into bilateral is as follows.

It is very interesting that the bilateral can be written as bilateral third-order mock theta function _{.}

We use Baileyβs bilateral transformation [15, 5.20(ii), page 137]:

Letting , and setting , , and in (3.4), we get

#### 4. Another Bilateral Transformation

Slater [15, (5.4.3), page 129] gave the following transformation formula, and we have taken :

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

In the transformation it is interesting that the *c*βs are absent in the series on the left side of (4.1). This gives us the freedom to choose the *c*βs in a convenient way.

Letting and setting, , , and in (4.1), so and , to get

By choosing suitably we can have different expansion identities. Moreover (4.2) can be seen as a generalization of (3.3).

#### 5. Mild Generalization of

We define the bilateral generalized function as

For , , reduce to

Now

So is an -function.

Being -function it has unified properties of -functions. For example, one has the following.

(i) The inverse operator of -differentiation is related to -integration as

See Jackson [16].

(ii) , where is a nonnegative integer.

#### 6. Behaviour of outside the Unit Circle

By definition (1.1)

Replacing by and writing for [10],

which is a -function.

#### 7. Lambert Series Expansion for

For the double series expansion, we first require the generalized Lambert series expansion for _{.}

By Entry 12.4.5, of Ramanujanβs Lost Notebook [17, page 277], Hikami [10] noted that

where

There is a slight misprint in the definition in Hikamiβs paper [10] which has been corrected and Gordon and McIntosh have also pointed out in their survey [8].

In [8] the Lambert series expansion for is

#### 8. as a Coefficient of of a -Function

In the following theorem of Hickerson [13, Theorem , page 645],

let , and then put to get

For , and and not an integral power of , let

Theorem 8.1. *Let be fixed with . Then is the coefficient of in the Laurent series expansion of in the annulus .*

*Proof. *By (7.3)
dividing by gives the theorem.

#### 9. An Identity for

Theorem 9.1. *If and z is neither zero nor an integral power of , then
**
Define
**
The scheme will be first to show that satisfies the functional relation:
**
One considers the poles of and and shows that the residue of at these poles is zero. So is analytic at these points. One then shows that the coefficients of in and are zero and equating the coefficient of in (9.4) one has the theorem.*

*Proof. *We show that
We shall show that each of , , and satisfies the functional equation:
and so
We employ (2.11) on the right-hand side to get
We now take :
Writing for on the right-hand side we have
Similarly only writing for we have
Hence the functional equation (9.4) is proved.

Obviously and are meromorphic for . has simple poles at and has simple poles at Hence is meromorphic for with, at most, simple poles at

Taking in (9.2), we calculate the residue of at the point :

For the residue of at take , , , in (2.16) to get
So the residue of at is

Now we calculate the residue at :

and for the residue of at taking , , , and in (2.16), so
Hence the residue of at is Hence is analytic at .

Since satisfies (9.4), so is analytic at all points of the form and hence for all

We now apply (2.20) with and and replaced by to get

where is the coefficient of in the Laurent expansion of .

Now for , by Theorem 8.1, the coefficient of in is

For such , if and only if .

That is,

Hence by (2.15)
So
If , then is either or ; so coefficient of in is 0. Similarly the coefficient of in is 0 and so the coefficient of in is

Hence by (9.17), we have

which gives the theorem.

#### 10. Double Series Expansion

Now we derive the double series expansion for . We shall use the Bailey pair method, as used by Andrews [18] for fifth and seventh-order mock theta functions and by Andrews and Hickerson [5] for sixth-order mock theta functions.

We define Bailey pair.

Two sequences and , , form a Bailey pair relative to a number if

for all .

Corollary 10.1 (see [5, Corollary. 2.1, page 70]). *If and form a Bailey pair relative to , then
**
provided that both sums converge absolutely.*

We state the theorem of Andrews and Hickerson [5, Theorem , pages 72-73].

Let , , , and be complex numbers with , , , , and none , , , of the form with . For , define

Then the sequences and form a Bailey pair relative to .

Letting and then taking , , in (10.3), we get

Now letting and then setting , , in (10.2) we get

Taking and for and , respectively, in (10.5) and using the definition of , we get

or

which is the double series expansion for .

This double series expansion can be used to get more properties of _{.}