Research Article | Open Access
A Mock Theta Function of Second Order
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 Ramanujan’s last letter to Hardy, dated January 12, 1920, to be specific [1, 2]. Ramanujan gave a list of seventeen functions which he called “mock theta functions.” 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 Ramanujan’s fifth and seventh-order mock theta functions  Watson  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 “Lost” Notebook. In the lost notebook were six more mock theta functions and linear relation between them. Andrews and Hickerson  called these mock theta functions of sixth-order and proved the identities.
In the “Lost” 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 .
Gordon and McIntosh listed eight functions in their eighth-order paper , but later, in their survey paper , 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  considered three second-order mock theta functions and gave transformation formulas for them. Hikami  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 “2nd” order.
He further showed that is a sum of two mock theta functions and where is of second-order and is Ramanujan’s 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  have shown that is essentially the odd part of the second-order mock theta function , which appears as in Andrews’ paper on Mordell integrals and Ramanujan’s lost notebook  and also in McIntosh paper on second-order mock theta functions . In particular,
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 Bailey’s 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 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
Ramanujan’s 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 Jacobi’s 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  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
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 Bailey’s 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’s are absent in the series on the left side of (4.1). This gives us the freedom to choose the c’s in a convenient way.
Letting and setting, , , and in (4.1), so and , to get
5. Mild Generalization of
We define the bilateral generalized function as
For , , reduce to
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 .
(ii) , where is a nonnegative integer.
6. Behaviour of outside the Unit Circle
By definition (1.1)
Replacing by and writing for ,
which is a -function.
7. Lambert Series Expansion for
For the double series expansion, we first require the generalized Lambert series expansion for .
In  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:
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 .
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  for fifth and seventh-order mock theta functions and by Andrews and Hickerson  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
which is the double series expansion for .
This double series expansion can be used to get more properties of .
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Copyright © 2009 Bhaskar Srivastava. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.