International Journal of Mathematics and Mathematical Sciences

International Journal of Mathematics and Mathematical Sciences / 2009 / Article

Research Article | Open Access

Volume 2009 |Article ID 978425 | https://doi.org/10.1155/2009/978425

Bhaskar Srivastava, "A Mock Theta Function of Second Order", International Journal of Mathematics and Mathematical Sciences, vol. 2009, Article ID 978425, 15 pages, 2009. https://doi.org/10.1155/2009/978425

A Mock Theta Function of Second Order

Academic Editor: Rodica Costin
Received10 Sep 2009
Accepted31 Dec 2009
Published22 Apr 2010

Abstract

We consider the second-order mock theta function 𝒟5 (𝑞), 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 𝒟5 (𝑞) outside the unit circle is a theta function and also write 1(𝑞) as a coefficient of 𝑧0 of a theta series. First writing 1(𝑞) as a coefficient of a theta function, we prove an identity for 1(𝑞).

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 [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 “Lost” 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 “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 [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: 𝒟5(𝑞)=𝑛=0𝑞𝑛(𝑞;𝑞)𝑛𝑞;𝑞2𝑛+1=1(1.1)(𝑞;𝑞2)2𝑛=0𝑞;𝑞22𝑛𝑞2𝑛(1.2) and proved that 𝒟5(𝑞) is a mock theta function and called it of “2nd” order.

He further showed that 𝒟5(𝑞) is a sum of two mock theta functions 1(𝑞) and 𝜔(𝑞) where 1(𝑞) is of second-order and 𝜔(𝑞) is Ramanujan’s mock theta function of third-order. This 𝒟5(𝑞) will be the basis of our study in this paper.

Before we begin with the study of 𝒟5(𝑞) and 1(𝑞) it will be appropriate to mention the work done earlier.

Gordon and McIntosh in their survey paper [8] have shown that 1(𝑞) is essentially the odd part of the second-order mock theta function 𝐵(𝑞), which appears as 𝛽(𝑞) in Andrews’ paper on Mordell integrals and Ramanujan’s lost notebook [11] and also in McIntosh paper on second-order mock theta functions [9]. In particular,

1𝑞2=𝐵(𝑞)𝐵(𝑞)4𝑞,(1.3) where

𝐵(𝑞)=𝑛=0𝑞𝑛(𝑛+1)𝑞2;𝑞2𝑛𝑞;𝑞22𝑛+1=𝑛=0𝑞𝑛𝑞;𝑞2𝑛𝑞;𝑞2𝑛+1.(1.4) Since the even part of 𝐵(𝑞) is the ordinary theta function

𝐵(𝑞)+𝐵(𝑞)2=𝑞4;𝑞4𝑞2;𝑞24,(1.5) it follows that the odd part and 1(𝑞) are second-order mock theta functions. Thus 𝒟5(𝑞) 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 𝒟5(𝑞) as a bilateral q-series and show that it is also a sum of the second-order mock theta function 𝒟5(𝑞) and the third-order mock theta function 𝜔(𝑞). By using Bailey’s transformation we have the interesting result that the bilateral 𝒟5,𝑐(𝑞) is the same as the bilateral 𝜔𝑐(𝑞).

In Section 4, using bilateral transformation of Slater, we write 𝒟5,𝑐(𝑞) as a bilateral series 2𝜓2 series with a free parameter c.

In Section 5, a mild generalization 𝒟5,𝑐(𝑧,𝛼) of 𝒟5,𝑐(𝑞) is given and we show that this generalized function is a 𝐹𝑞-function.

In Section 6 we show that 𝒟5(𝑞), outside the unit circle |𝑞|=1, is a theta function.

In Section 7 we state a generalized Lambert Series expansion for 1(𝑞) as given in [8].

In Section 8 we show that 1(𝑞) is a coefficient of 𝑧0 of a theta function.

In Section 9 we prove an identity for 1(𝑞) using 1(𝑞) as a coefficient of 𝑧0 of a theta function.

In Section 10 a double series expansion for 1(𝑞) is obtained by using Bailey pair method.

2. Basic Preliminaries

We first introduce some standard notation.

If 𝑞 and 𝑎 are complex numbers with |𝑞|<1 and 𝑛 is a nonnegative integer, then

(𝑎)0=(𝑎;𝑞)0=1,(𝑎)𝑛=(𝑎;𝑞)𝑛=𝑛1𝑘=01𝑎𝑞𝑘,(𝑎)=(𝑎;𝑞)=𝑘=01𝑎𝑞𝑘,𝑎1,,𝑎𝑚𝑛=𝑎1,,𝑎𝑚;𝑞𝑛=𝑎1;𝑞𝑛𝑎,,𝑚;𝑞𝑛.(2.1) Ramanujan’s mock theta function of third-order 𝜔(𝑞) and 𝜈(𝑞) is

𝜔(𝑞)=𝑛=0𝑞2𝑛(𝑛+1)𝑞;𝑞22𝑛+1,(2.2)𝜈(𝑞)=𝑛=0𝑞𝑛(𝑛+1)𝑞;𝑞2𝑛+1,(2.3)𝜑(𝑞)=𝑛=𝑞𝑛2=𝑞;𝑞22𝑞2;𝑞2=(𝑞;𝑞)(𝑞;𝑞).(2.4) We will use the following notations for 𝜃-functions.

Definition 2.1. If |𝑞|<1 and 𝑥0, then 𝑞𝑗(𝑥,𝑞)=𝑥,𝑥,𝑞;𝑞.(2.5) If 𝑚 is a positive integer and 𝑎 is an integer, 𝐽𝑎,𝑚=𝑗(𝑞𝑎,𝑞𝑚),(2.6)𝐽𝑎,𝑚=𝑗(𝑞𝑎,𝑞𝑚𝐽),(2.7)𝑚𝑞=𝑗𝑚,𝑞3𝑚=(𝑞𝑚;𝑞𝑚)𝑗𝑞,(2.8)𝑥𝑥,𝑞=𝑗(𝑥,𝑞),(2.9)𝑗(𝑥,𝑞)=𝑥𝑗1,𝑞,(2.10)𝑗(𝑞𝑛𝑥,𝑞)=(1)𝑛𝑞𝑛(𝑛1)/2𝑥𝑛𝑗(𝑥,𝑞),if𝑛isaninteger.(2.11) By Jacobi’s triple product identity [12, page 282] 𝑗(𝑥,𝑞)=𝑛=(1)𝑛𝑞𝑛(𝑛1)/2𝑥𝑛.(2.12)

2.1. More Definitions

If 𝑧 is a complex number with |𝑧|1, then

𝜀(𝑧)=1if|𝑧|<1,1if|𝑧|>1.(2.13)

If s is an integer, then

sg(𝑠)=1if𝑠0,1if𝑠<0.(2.14)

Using these definitions,

11𝑧=𝜀(𝑧)𝑠=sg(𝑠)=𝜀(𝑧)𝑧𝑠.(2.15) We shall use the following theorems.

Theorem 2.2 (see [13, Theorem 1.3, page 644]). Let 𝑞 be fixed, 0<|𝑞|<1. Let 𝑎,𝑏, and 𝑚 be fixed integers with 𝑏0 and 𝑚1. Define 1𝐹(𝑧)=𝑗𝑞𝑎𝑧𝑏,𝑞𝑚.(2.16) Then 𝐹 is meromorphic for 𝑧0, with simple poles at all points 𝑧0 such that 𝑧𝑏0=𝑞𝑘𝑚𝑎 for some integer 𝑘. The residue of 𝐹(𝑧) at such a point 𝑧0 is (1)𝑘+1𝑞𝑚𝑘(𝑘1)/2𝑧0𝑏𝐽3𝑚.(2.17)

Theorem 2.3 (see [13, Theorem 1.8(a), page 647]). Suppose that 𝐹(𝑧)=𝑟𝐹𝑟𝑧𝑟(2.18) for all 𝑧0 and that 𝐹(𝑧) satisfies 𝐹(𝑞𝑧)=𝐶𝑧𝑛𝐹(𝑧),(2.19) where 0<|𝑞|<1 and 𝐶0. Then 𝐹(𝑧)=𝑛1𝑟=0𝐹𝑟𝑧𝑟𝑗𝐶1𝑞𝑟𝑧𝑛,𝑞𝑛.(2.20) Truesdell [14] calls the functions which satisfy the difference equation 𝜕𝜕𝑧𝐹(𝑧,𝛼)=𝐹(𝑧,𝛼+1)(2.21) as 𝐹-function. He unified the study of these 𝐹-functions.
The functions which satisfy the 𝑞-analogue of the difference equation
𝐷𝑞,𝑧𝐹(𝑧,𝛼)=𝐹(𝑧,𝛼+1),(2.22) where 𝑧𝐷𝑞,𝑧𝐹(𝑧,𝛼)=𝐹(𝑧,𝛼)𝐹(𝑧𝑞,𝛼)(2.23) are called 𝐹𝑞-functions.

3. Bilateral 𝒟5(𝑞) as a Sum of Two Mock Theta Functions of Different Orders

(i) We shall denote the bilateral of 𝒟5(𝑞) by 𝒟5,𝑐(𝑞). We define it as

𝑞;𝑞22𝒟5,𝑐(𝑞)=𝑛=𝑞;𝑞22𝑛𝑞2𝑛.(3.1) Now

𝑞;𝑞22𝒟5,𝑐(𝑞)=𝑛=𝑞;𝑞22𝑛𝑞2𝑛=𝑛=0𝑞;𝑞22𝑛𝑞2𝑛+𝑛=1𝑞;𝑞22𝑛𝑞2𝑛=𝑛=0𝑞;𝑞22𝑛𝑞2𝑛+𝑛=0𝑞2𝑛2+2𝑛𝑞;𝑞22𝑛+1,(3.2) and we use (1.2) in the first summation and (2.2) in the second summation, to write

𝑞;𝑞22𝒟5,𝑐(𝑞)=𝑞;𝑞22𝒟5(𝑞)+𝜔(𝑞).(3.3)

Thus 𝒟5,𝑐(𝑞) is a sum of a second-order mock theta function and a third-order mock theta function.

(ii) Transformation of Bilateral 𝒟5,𝑐(𝑞) into bilateral 𝜔𝑐(𝑞) is as follows.

It is very interesting that the bilateral 𝒟5,𝑐(𝑞) can be written as bilateral third-order mock theta function 𝜔𝑐(𝑞).

We use Bailey’s bilateral transformation [15, 5.20(ii), page 137]:

2𝜓2=𝑎,𝑏𝑐,𝑑;𝑞,𝑧(𝑎𝑧,𝑏𝑧,𝑐𝑞/𝑎𝑏𝑧,𝑑𝑞/𝑎𝑏𝑧;𝑞)(𝑞/𝑎,𝑞/𝑏,𝑐,𝑑;𝑞)×2𝜓2𝑎𝑏𝑧/𝑐,𝑎𝑏𝑧/𝑑𝑎𝑧,𝑏𝑧;𝑞,𝑐𝑑/𝑎𝑏𝑧.(3.4) Letting 𝑞𝑞2, and setting 𝑎=𝑏=𝑞, 𝑐=𝑑=0, and 𝑧=𝑞2 in (3.4), we get

𝑞;𝑞22𝒟5,𝑐𝑞(𝑞)=3,𝑞3;𝑞2𝑞,𝑞;𝑞2𝑛=𝑞2𝑛2+2𝑛𝑞3;𝑞22𝑛=𝑛=𝑞2𝑛2+2𝑛𝑞;𝑞22𝑛+1=𝜔𝑐(𝑞).(3.5)

4. Another Bilateral Transformation

Slater [15, (5.4.3), page 129] gave the following transformation formula, and we have taken 𝑟=2:

𝑏1,𝑏2,𝑞/𝑎1,𝑞/𝑎2,𝑑𝑧,𝑞/𝑑𝑧;𝑞𝑐1,𝑐2,𝑞/𝑐1,𝑞/𝑐2;𝑞2𝜓2𝑎1,𝑎2𝑏1,𝑏2=𝑞;𝑞,𝑧𝑐1𝑐1/𝑎1,𝑐1/𝑎2,𝑞𝑏1/𝑐1,𝑞𝑏2/𝑐1,𝑑𝑐1𝑧/𝑞,𝑞2/𝑑𝑐1𝑧;𝑞𝑐1,𝑞/𝑐1,𝑐1/𝑐2,𝑞𝑐2/𝑐1;𝑞2𝜓2𝑞𝑎1/𝑐1,𝑞𝑎2/𝑐1𝑞𝑏1/𝑐1,𝑞𝑏2/𝑐1𝑐;𝑞,𝑧+idem1;𝑐2,(4.1) where 𝑑=𝑎1𝑎2/𝑐1𝑐2, |𝑏1𝑏2/𝑎1𝑎2|<|𝑧|<1, and idem(𝑐1;𝑐2) after the expression means that the preceding expression is repeated with 𝑐1 and 𝑐2 interchanged.

In the transformation it is interesting that the c’s are absent in the 2𝜓2 series on the left side of (4.1). This gives us the freedom to choose the c’s in a convenient way.

Letting 𝑞𝑞2 and setting, 𝑎1=𝑎2=𝑞, 𝑏1=𝑏2=0, and 𝑧=𝑞2 in (4.1), so 𝑑=𝑞2/𝑐1𝑐2 and 0<|𝑧|<1, to get

𝑞;𝑞24𝑞4/𝑐1𝑐2;𝑞2𝑐1𝑐2/𝑞2;𝑞2𝑐2;𝑞2𝑞2/𝑐2;𝑞2𝒟5,𝑐=𝑞(𝑞)2𝑐1𝑐1/𝑞;𝑞22𝑞2/𝑐2;𝑞2𝑐2;𝑞2𝑐1/𝑐2;𝑞2𝑞2𝑐2/𝑐1;𝑞2𝑛=𝑞3/𝑐1;𝑞22𝑛𝑞2𝑛𝑐+idem1;𝑐2.(4.2) By choosing 𝑐1 suitably we can have different expansion identities. Moreover (4.2) can be seen as a generalization of (3.3).

5. Mild Generalization of 𝒟5,𝑐(𝑞)

We define the bilateral generalized function 𝒟5,𝑐(𝑧,𝛼) as

𝑞;𝑞22𝒟5,𝑐1(𝑧,𝛼)=(𝑧)𝑛=𝑞;𝑞22𝑛(𝑧)𝑛𝑞𝑛𝛼+𝑛.(5.1) For 𝛼=1, 𝑧=0, 𝒟5,𝑐(𝑧,𝛼) reduce to 𝒟5,𝑐(𝑞).

Now

𝐷𝑞,𝑧𝒟5,𝑐=1(𝑧,𝛼)𝑧𝒟5,𝑐(𝑧,𝛼)𝒟5,𝑐=1(𝑧𝑞,𝛼)𝑧𝑞;𝑞221(𝑧)𝑛=𝑞;𝑞22𝑛(𝑧)𝑛𝑞𝑛𝛼+𝑛1(𝑧𝑞)𝑛=𝑞;𝑞22𝑛(𝑧𝑞)𝑛𝑞𝑛𝛼+𝑛=1𝑧𝑞;𝑞221(𝑧)𝑛=𝑞;𝑞22𝑛(𝑧)𝑛𝑞𝑛𝛼+𝑛(1(1𝑧𝑞𝑛=1))𝑞;𝑞22(𝑧)𝑛=𝑞;𝑞22𝑛(𝑧)𝑛𝑞𝑛𝛼+2𝑛=𝒟5,𝑐(𝑧,𝛼+1).(5.2) So 𝒟5,𝑐(𝑧,𝛼+1) is an 𝐹𝑞-function.

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

(i) The inverse operator 𝐷1𝑞,𝑥 of 𝑞-differentiation is related to 𝑞-integration as

𝐷1𝑞,𝑥𝜙(𝑥)=(1𝑞)1𝜙(𝑥)𝑑𝑞𝑥.(5.3) See Jackson [16].

(ii) 𝐷𝑛𝑞,𝑧𝐹𝑞(𝑧,𝛼)=𝐹𝑞(𝑧,𝛼+𝑛), where 𝑛 is a nonnegative integer.

6. Behaviour of 𝒟5(𝑞) outside the Unit Circle

By definition (1.1)

𝒟5(𝑞)=𝑛=0(𝑞;𝑞)𝑛𝑞;𝑞2𝑛+1𝑞𝑛.(6.1) Replacing 𝑞 by 1/𝑞 and writing 𝒟5(𝑞) for 𝒟5(1/𝑞) [10],

𝒟5(𝑞)=𝑛=0(1)𝑛𝑞(𝑛2+𝑛)/2(𝑞;𝑞)𝑛𝑞;𝑞2𝑛+1=1𝑞2+𝑞6𝑞12+𝑞20𝑞30=+𝑛=0(1)𝑛𝑞𝑛2+𝑛,(6.2) which is a 𝜃-function.

7. Lambert Series Expansion for 1(𝑞)

For the double series expansion, we first require the generalized Lambert series expansion for 1(𝑞).

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

𝒟5(𝑞)=21(𝑞)(𝑞;𝑞)2𝜔(𝑞),(7.1) where

1(𝑞)=𝑛=0(𝑞;𝑞)2𝑛𝑞;𝑞22𝑛+1𝑞𝑛.(7.2) There is a slight misprint in the definition 1(𝑞) in Hikami’s 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 1(𝑞) is

1(𝑞)=𝑛=0(𝑞;𝑞)2𝑛𝑞;𝑞22𝑛+1𝑞𝑛=1𝜃4(0,𝑞)𝑛=0(1)𝑛𝑞𝑛(𝑛+2)1𝑞2𝑛+1=12𝜃4(0,𝑞)𝑛=(1)𝑛𝑞𝑛(𝑛+2)1𝑞2𝑛+1.(7.3)

8. 1(𝑞) as a Coefficient of 𝑧0 of a 𝜃-Function

In the following theorem of Hickerson [13, Theorem 1.4, page 645],

𝑟=𝑥𝑟1𝑞𝑟𝑦=𝐽31𝑗(𝑥𝑦,𝑞)𝑗(𝑥,𝑞)𝑗(𝑦,𝑞)(8.1) let 𝑞𝑞2, and then put 𝑦=𝑞, to get

𝑟=𝑥𝑟1𝑞2𝑟+1=𝐽32𝑗𝑞𝑥,𝑞2𝑗𝑥,𝑞2𝑗𝑞,𝑞2.(8.2) For |𝑞|<1, and 𝑧0 and not an integral power of 𝑞, let

𝐴1(𝑧)=2𝜃4𝐽(0,𝑞)32𝑗𝑞𝑧,𝑞2𝑗𝑧,𝑞2𝑗𝑞,𝑞2𝑗𝑧𝑞,𝑞2.(8.3)

Theorem 8.1. Let 𝑞 be fixed with 0<|𝑞|<1. Then 1(𝑞) is the coefficient of 𝑧0 in the Laurent series expansion of 𝐴(𝑧) in the annulus |𝑞|<|𝑧|<1.

Proof. By (7.3) 2𝜃4(0,𝑞)1(𝑞)=𝑛=(1)𝑛𝑞𝑛(𝑛+2)1𝑞2𝑛+1=coecientof𝑧0in𝑛=𝑧𝑛1𝑞2𝑛+1𝑠=(1)𝑠𝑞𝑠2+𝑠𝑧𝑞𝑠=coecientof𝑧0𝐽in32𝑗𝑞𝑧,𝑞2𝑗𝑧,𝑞2𝑗𝑞,𝑞2𝑗𝑧𝑞,𝑞2(8.4) dividing by 2𝜃4(0,𝑞) gives the theorem.

9. An Identity for 1(𝑞)

Theorem 9.1. If 0<|𝑞|<1 and z is neither zero nor an integral power of 𝑞, then 𝐴(𝑧,𝑞)=𝑗𝑧,𝑞211(𝑞)2𝑟=(1)𝑟𝑞𝑟2+3𝑟1𝑧𝑟+21𝑞2𝑟+2𝑧12𝑟=(1)𝑟𝑞𝑟2+3𝑟+1𝑧𝑟11𝑞2𝑟+2𝑧1.(9.1) Define 1𝐿(𝑧)=2𝑟=(1)𝑟𝑞𝑟2+3𝑟1𝑧𝑟+21𝑞2𝑟+2𝑧1(9.2)𝑀(𝑧)=2𝑟=(1)𝑟𝑞𝑟2+3𝑟+1𝑧𝑟11𝑞2𝑟+2𝑧1𝐹,(9.3)(𝑧)=𝐴(𝑧)+𝐿(𝑧)+𝑀(𝑧).(9.4) The scheme will be first to show that 𝐹(𝑧) satisfies the functional relation: 𝐹𝑞2𝑧=𝑧1𝐹(𝑧).(9.5) One considers the poles of 𝐿(𝑧) and 𝑀(𝑧) and shows that the residue of 𝐹(𝑧) at these poles is zero. So 𝐹(𝑧) is analytic at these points. One then shows that the coefficients of 𝑧0 in 𝐿(𝑧) and 𝑀(𝑧) are zero and equating the coefficient of 𝑧0 in (9.4) one has the theorem.

Proof. We show that 𝐹𝑞2𝑧=𝑧1𝐹(𝑧).(9.6) We shall show that each of 𝐴(𝑧), 𝐿(𝑧), and 𝑀(𝑧) satisfies the functional equation: 𝐴1(𝑧)=2(𝑞;𝑞)(𝑞;𝑞)𝐽32𝑗𝑞𝑧,𝑞2𝑗𝑧,𝑞2𝑗𝑞,𝑞2𝑗𝑧𝑞,𝑞2,(9.7) and so 𝐴𝑞2𝑧=12(𝑞;𝑞)(𝑞;𝑞)𝐽32𝑗𝑞3𝑧,𝑞2𝑗𝑞2𝑧,𝑞2𝑗𝑞,𝑞2𝑗𝑧𝑞,𝑞2.(9.8) We employ (2.11) on the right-hand side to get 𝐴𝑞2𝑧=12(𝑞;𝑞)(𝑞;𝑞)𝐽32(1)𝑧1𝑞1𝑗𝑧𝑞,𝑞2(1)𝑞𝑧1(1)𝑧1𝑗𝑧,𝑞2𝑗𝑞,𝑞2𝑗𝑧𝑞,𝑞2𝐴𝑞2𝑧=𝑧1𝐴(𝑧).(9.9) We now take 𝐿(𝑧): 𝐿𝑞2𝑧=12𝑟=(1)𝑟𝑞𝑟2+3𝑟1𝑧𝑞2𝑟+21𝑞2𝑟+2𝑧𝑞2.(9.10) Writing 𝑟1 for 𝑟on the right-hand side we have 𝐿𝑞2𝑧=𝑧1𝐿(𝑧).(9.11) Similarly only writing 𝑟+1 for 𝑟 we have 𝑀𝑞2𝑧=𝑧1𝑀(𝑧).(9.12) Hence the functional equation (9.4) is proved.
Obviously 𝐿(𝑧) and 𝑀(𝑧) are meromorphic for 𝑧0. 𝐿(𝑧) has simple poles at 𝑧=𝑞2𝑘2 and 𝑀(𝑧) has simple poles at 𝑧=𝑞2𝑘+2. Hence 𝐹(𝑧) is meromorphic for 𝑧0 with, at most, simple poles at 𝑧=𝑞2𝑘±2.
Taking 𝑟=0 in (9.2), we calculate the residue of 𝐿(𝑧) at the point 𝑧=1/𝑞2:
Residueof𝐿(𝑧)=lim𝑧1/𝑞2121𝑧𝑞2𝑧2𝑞1𝑧1/𝑞2𝑞2=12𝑞5.(9.13) For the residue of 𝐴(𝑧) at 𝑧=1/𝑞2, take 𝑏=1, 𝑘=1, 𝑚=2, 𝑎=0 in (2.16) to get 1Residueof𝐴(𝑧)=2(𝑞;𝑞)(𝑞;𝑞)𝐽32𝑗1/𝑞,𝑞2𝑗1/𝑞3,𝑞2𝑗𝑞,𝑞21𝐽32=12(𝑞;𝑞)(𝑞;𝑞)𝑗𝑞,𝑞2𝑗𝑞3,𝑞2𝑞4𝑗𝑞,𝑞2=12(𝑞;𝑞)(𝑞;𝑞)1𝑞4𝑞3;𝑞21𝑞;𝑞2𝑞2;𝑞2=12(𝑞;𝑞)(𝑞;𝑞)1𝑞4(1(1/𝑞))(1𝑞)𝑞;𝑞2𝑞;𝑞2𝑞2;𝑞21=2(𝑞;𝑞)(𝑞;𝑞)1𝑞5(𝑞;𝑞)(𝑞;𝑞)1=2𝑞5.(9.14) So the residue of 𝐹(𝑧) at 𝑧=1/𝑞2 is (1/2)𝑞5+0+(1/2)𝑞5=0.
Now we calculate the residue at 𝑧=𝑞2:
Residueof𝑀(𝑧)=lim𝑧𝑞212𝑧𝑞2𝑞𝑧11𝑞2𝑧1=lim𝑧𝑞212𝑧𝑞2𝑞𝑧𝑞2=𝑞2,(9.15) and for the residue of 𝐴(𝑧) at 𝑧=𝑞2, taking 𝑏=1, 𝑘=1, 𝑚=2, and 𝑎=0 in (2.16), so 1Residueof𝐴(𝑧)=2(𝑞;𝑞)(𝑞;𝑞)𝐽32𝑗𝑞3,𝑞2𝑗𝑞,𝑞2𝑗𝑞,𝑞2𝑞2𝐽32=12(𝑞;𝑞)(𝑞;𝑞)𝑗𝑞3,𝑞2𝑞2=12(𝑞;𝑞)(𝑞;𝑞)𝑞3;𝑞21𝑞;𝑞2𝑞2;𝑞2𝑞2=12(𝑞;𝑞)(𝑞;𝑞)(1(1/𝑞))(1𝑞)𝑞;𝑞2𝑞;𝑞2𝑞2;𝑞2𝑞21=2𝑞.(9.16) Hence the residue of 𝐹(𝑧) at 𝑧=𝑞2 is 0+(1/2)𝑞(1/2)𝑞=0. Hence 𝐹(𝑧) is analytic at 𝑧=𝑞2.
Since 𝐹(𝑧) satisfies (9.4), so 𝐹(𝑧) is analytic at all points of the form 𝑧=𝑞2𝑘±2 and hence for all 𝑧0.
We now apply (2.20) with 𝑛=1 and 𝑐=1 and 𝑞 replaced by 𝑞2 to get
𝐹(𝑧)=𝐹0𝑗𝑧,𝑞2,(9.17) where 𝐹0 is the coefficient of 𝑧0 in the Laurent expansion of 𝐹(𝑧),𝑧0.
Now for |𝑞|<|𝑧|<1, by Theorem 8.1, the coefficient of 𝑧0 in 𝐴(𝑧) is 1(𝑞).
For such 𝑧, |𝑞2𝑟+2𝑧|<1 if and only if 𝑟0.
That is,
𝜀𝑞2𝑟+2𝑧=sg(𝑟).(9.18) Hence by (2.15) 11𝑞2𝑟+2𝑧=sg(𝑟)𝑟=sg(𝑟)=sg(𝑠)𝑞(2𝑟+2)𝑠𝑧𝑠.(9.19) So 1𝐿(𝑧)=2sg(𝑟)=sg(𝑠)sg(𝑟)(1)𝑟𝑞𝑟2+3𝑟1+(2𝑟+2)𝑠𝑧𝑟+2+𝑠.(9.20) If sg(𝑟)=sg(𝑠), then 𝑟+𝑠+2 is either 1 or 1; so coefficient of 𝑧0 in 𝐿(𝑧) is 0. Similarly the coefficient of 𝑧0 in 𝑀(𝑧) is 0 and so the coefficient of 𝑧0 in 𝐹(𝑧) is 1(𝑞).
Hence by (9.17), we have
𝐹(𝑧)=1(𝑞)𝑗𝑧,𝑞2,(9.21) which gives the theorem.

10. Double Series Expansion

Now we derive the double series expansion for 1(𝑞). 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 {𝛼𝑛} and {𝛽𝑛}, 𝑛0, form a Bailey pair relative to a number 𝑎 if

𝛽𝑛=𝑛𝑟=0𝛼𝑟(𝑞)𝑛𝑟(𝑎𝑞)𝑛+𝑟,(10.1) for all 𝑛0.

Corollary 10.1 (see [5, Corollary. 2.1, page 70]). If {𝛼𝑛} and {𝛽𝑛} form a Bailey pair relative to 𝑎, then 𝑛=0𝜌1𝑛𝜌2𝑛𝑎𝑞/𝜌1𝜌2𝑛𝛼𝑛𝑎𝑞/𝜌1𝑛𝑎𝑞/𝜌2𝑛=(𝑎𝑞)𝑎𝑞/𝜌1𝜌2𝑎𝑞/𝜌1𝑎𝑞/𝜌2𝑛=0𝜌1𝑛𝜌2𝑛𝑎𝑞𝜌1𝜌2𝑛𝛽𝑛(10.2) provided that both sums converge absolutely.

We state the theorem of Andrews and Hickerson [5, Theorem 2.3, pages 72-73].

Let 𝑎, 𝑏, 𝑐, and 𝑞 be complex numbers with 𝑎1, 𝑏0, 𝑐0, 𝑞0, and none 𝑎/𝑏, 𝑎/𝑐, 𝑞𝑏, 𝑞𝑐 of the form 𝑞𝑘 with 𝑘0. For 𝑛0, define

𝐴𝑛𝑞(𝑎,𝑏,𝑐,𝑞)=𝑛2(𝑏𝑐)𝑛1𝑎𝑞2𝑛(𝑎/𝑏)𝑛(𝑎/𝑐)𝑛(1𝑎)(𝑞𝑏)𝑛(𝑞𝑐)𝑛×𝑛𝑗=0(1)𝑗1𝑎𝑞2𝑗1(𝑎)𝑗1(𝑏)𝑗(𝑐)𝑗𝑞(𝑗2)(𝑏𝑐)𝑗(𝑞)𝑗(𝑎/𝑏)𝑗(𝑎/𝑐)𝑗,𝐵𝑛1(𝑎,𝑏,𝑐,𝑞)=(𝑞𝑏)𝑛(𝑞𝑐)𝑛.(10.3) Then the sequences {𝐴𝑛(𝑎,𝑏,𝑐,𝑞)} and {𝐵𝑛(𝑎,𝑏,𝑐,𝑞)} form a Bailey pair relative to 𝑎.

Letting 𝑞𝑞2 and then taking 𝑎=𝑞2, 𝑏=𝑐=𝑞, in (10.3), we get

𝐴𝑛𝑞2,𝑞,𝑞2=𝑞2𝑛2+2𝑛1𝑞4𝑛+2𝑞;𝑞22𝑛1𝑞2𝑞3;𝑞22𝑛×𝑛𝑗=0(1)𝑗1𝑞4𝑗𝑞2;𝑞2𝑗1𝑞;𝑞22𝑗𝑞𝑗2+𝑗𝑞2;𝑞2𝑗𝑞;𝑞22𝑗=(1𝑞)21+𝑞2𝑛+1𝑞2𝑛2+2𝑛1𝑞21𝑞2𝑛+1𝑛𝑗=0(1)𝑗𝑞𝑗2𝑗1+𝑞2𝑗=(1𝑞)21+𝑞2𝑛+1𝑞2𝑛2+2𝑛1𝑞21𝑞2𝑛+11+𝑛𝑗=𝑛(1)𝑗𝑞𝑗2𝑗,𝐵𝑛𝑞2,𝑞,𝑞,𝑞2=1𝑞3;𝑞22𝑛.(10.4) Now letting 𝑞𝑞2 and then setting 𝜌1=𝑞, 𝜌2=𝑞2, 𝑎=𝑞2 in (10.2) we get

𝑛=0𝑞𝑛𝑞;𝑞2𝑛𝑞3;𝑞2𝑛𝛼𝑛=𝑞;𝑞2𝑞4;𝑞2𝑞2;𝑞2𝑞3;𝑞2𝑛=0𝑞;𝑞2𝑛𝑞2;𝑞2𝑛𝑞𝑛𝛽𝑛.(10.5) Taking 𝐴𝑛 and 𝐵𝑛 for 𝛼𝑛 and 𝛽𝑛, respectively, in (10.5) and using the definition of 1(𝑞), we get

𝑞;𝑞2𝑞2;𝑞2𝑞;𝑞2𝑞2;𝑞21(𝑞)=𝑛=0𝑞2𝑛2+3𝑛1𝑞2𝑛+11+𝑛𝑗=𝑛(1)𝑗𝑞𝑗2𝑗(10.6) or

1(𝑞)=𝑞;𝑞2𝑞2;𝑞2𝑞;𝑞2𝑞2;𝑞2𝑛=0𝑞2𝑛2+3𝑛1𝑞2𝑛+11+𝑛𝑗=𝑛(1)𝑗𝑞𝑗2𝑗,(10.7) which is the double series expansion for 1(𝑞).

This double series expansion can be used to get more properties of 𝒟5(𝑞).

References

  1. S. Ramanujan, Collected Papers, Cambridge University Press, Cambridge, UK, 1927.
  2. S. Ramanujan, Collected Papers, Chelsea, New York, NY, USA, 1962.
  3. B. Gordon and R. J. McIntosh, “Modular transformations of Ramanujan's fifth and seventh order mock theta functions,” The Ramanujan Journal, vol. 7, pp. 193–222, 2003. View at: Google Scholar
  4. G. N. Watson, “The final problem: an account of the mock theta functions,” Journal of the London Mathematical Society, vol. 11, pp. 55–80, 1936. View at: Google Scholar
  5. G. E. Andrews and D. Hickerson, “Ramanujan's “lost” notebook. VII. The sixth order mock theta functions,” Advances in Mathematics, vol. 89, no. 1, pp. 60–105, 1991. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  6. Y.-S. Choi, “Tenth order mock theta functions in Ramanujan's lost notebook. IV,” Transactions of the American Mathematical Society, vol. 354, no. 2, pp. 705–733, 2002. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  7. B. Gordon and R. J. McIntosh, “Some eighth order mock theta functions,” Journal of the London Mathematical Society, vol. 62, no. 2, pp. 321–335, 2000. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  8. B. Gordon and R. J. McIntosh, “A survey of classical mock theta functions,” preprint. View at: Google Scholar
  9. R. J. McIntosh, “Second order mock theta functions,” Canadian Mathematical Bulletin, vol. 50, no. 2, pp. 284–290, 2007. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  10. K. Hikami, “Transformation formula of the “second” order mock theta function,” Letters in Mathematical Physics, vol. 75, no. 1, pp. 93–98, 2006. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  11. G. E. Andrews, “Mordell integrals and Ramanujan's lost notebook,” in Analytic Number Theory, Lecture Notes, vol. 899, pp. 10–48, Springer, Berlin, 1981. View at: Google Scholar
  12. G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Oxford University Press, London, UK, 4th edition, 1968.
  13. D. Hickerson, “A proof of the mock theta conjectures,” Inventiones Mathematicae, vol. 94, no. 3, pp. 639–660, 1988. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  14. C. Truesdell, An Essay Toward a Unified Theory of Special Functions, Annals of Mathematics Studies, no. 18, Princeton University Press, Princeton, NJ, USA, 1948. View at: MathSciNet
  15. G. Gasper and M. Rahman, Basic Hypergeometric Series, vol. 35 of Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, UK, 1990. View at: MathSciNet
  16. F. H. Jackson, “Basic integration,” Quart. J. Math. ( Oxford), (2), vol. 2, pp. 1–16, 1951. View at: Google Scholar
  17. G. E. Andrews and B. C. Berndt, Ramanujan's Lost Notebook, Part I, Springer, New York, NY, USA, 2005. View at: MathSciNet
  18. G. E. Andrews, “The fifth and seventh order mock theta functions,” Transactions of the American Mathematical Society, vol. 293, no. 1, pp. 113–134, 1986. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet

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.

Related articles

No related content is available yet for this article.
 PDF Download Citation Citation
 Download other formatsMore
 Order printed copiesOrder
Views1728
Downloads919
Citations

Related articles

No related content is available yet for this article.

Article of the Year Award: Outstanding research contributions of 2021, as selected by our Chief Editors. Read the winning articles.