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 | 15 pages | 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 π’Ÿ5 (π‘ž), 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 π’Ÿ5 (π‘ž) outside the unit circle is a theta function and also write β„Ž1(π‘ž) as a coefficient of 𝑧0 of a theta series. First writing β„Ž1(π‘ž) as a coefficient of a theta function, we prove an identity for β„Ž1(π‘ž).

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 β€œmock 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 β€œLost” 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 β€œLost” 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: π’Ÿ5(ξ“π‘ž)=βˆžπ‘›=0π‘žπ‘›(βˆ’π‘ž;π‘ž)π‘›ξ€·π‘ž;π‘ž2𝑛+1=1(1.1)(π‘ž;π‘ž2)2βˆžξ“βˆžπ‘›=0ξ€·π‘ž;π‘ž2ξ€Έ2π‘›π‘ž2𝑛(1.2) and proved that π’Ÿ5(π‘ž) is a mock theta function and called it of β€œ2nd” order.

He further showed that π’Ÿ5(π‘ž) is a sum of two mock theta functions β„Ž1(π‘ž) and πœ”(π‘ž) where β„Ž1(π‘ž) is of second-order and πœ”(π‘ž) is Ramanujan’s mock theta function of third-order. This π’Ÿ5(π‘ž) will be the basis of our study in this paper.

Before we begin with the study of π’Ÿ5(π‘ž) and β„Ž1(π‘ž) it will be appropriate to mention the work done earlier.

Gordon and McIntosh in their survey paper [8] have shown that β„Ž1(π‘ž) is essentially the odd part of the second-order mock theta function 𝐡(π‘ž), which appears as 𝛽(π‘ž) 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,

β„Ž1ξ€·π‘ž2ξ€Έ=𝐡(π‘ž)βˆ’π΅(βˆ’π‘ž)4π‘ž,(1.3) where

𝐡(π‘ž)=βˆžπ‘›=0π‘žπ‘›(𝑛+1)ξ€·βˆ’π‘ž2;π‘ž2ξ€Έπ‘›ξ€·π‘ž;π‘ž2ξ€Έ2𝑛+1=ξ“βˆžπ‘›=0π‘žπ‘›ξ€·βˆ’π‘ž;π‘ž2ξ€Έπ‘›ξ€·π‘ž;π‘ž2𝑛+1.(1.4) Since the even part of 𝐡(π‘ž) is the ordinary theta function

𝐡(π‘ž)+𝐡(βˆ’π‘ž)2=ξ€·π‘ž4;π‘ž4ξ€Έβˆžξ€·βˆ’π‘ž2;π‘ž2ξ€Έ4∞,(1.5) it follows that the odd part and β„Ž1(π‘ž) are second-order mock theta functions. Thus π’Ÿ5(π‘ž) 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 π’Ÿ5(π‘ž) as a bilateral q-series and show that it is also a sum of the second-order mock theta function π’Ÿ5(π‘ž) and the third-order mock theta function πœ”(π‘ž). By using Bailey’s transformation we have the interesting result that the bilateral π’Ÿ5,𝑐(π‘ž) is the same as the bilateral πœ”π‘(π‘ž).

In Section 4, using bilateral transformation of Slater, we write π’Ÿ5,𝑐(π‘ž) as a bilateral series 2πœ“2 series with a free parameter c.

In Section 5, a mild generalization π’Ÿ5,𝑐(𝑧,𝛼) of π’Ÿ5,𝑐(π‘ž) is given and we show that this generalized function is a πΉπ‘ž-function.

In Section 6 we show that π’Ÿ5(π‘ž), outside the unit circle |π‘ž|=1, is a theta function.

In Section 7 we state a generalized Lambert Series expansion for β„Ž1(π‘ž) as given in [8].

In Section 8 we show that β„Ž1(π‘ž) is a coefficient of 𝑧0 of a theta function.

In Section 9 we prove an identity for β„Ž1(π‘ž) using β„Ž1(π‘ž) as a coefficient of 𝑧0 of a theta function.

In Section 10 a double series expansion for β„Ž1(π‘ž) is obtained by using Bailey pair method.

2. Basic Preliminaries

We first introduce some standard notation.

If π‘ž and π‘Ž are complex numbers with |π‘ž|<1 and 𝑛 is a nonnegative integer, then

(π‘Ž)0=(π‘Ž;π‘ž)0=1,(π‘Ž)𝑛=(π‘Ž;π‘ž)𝑛=π‘›βˆ’1ξ‘π‘˜=0ξ€·1βˆ’π‘Žπ‘žπ‘˜ξ€Έ,(π‘Ž)∞=(π‘Ž;π‘ž)∞=βˆžξ‘π‘˜=0ξ€·1βˆ’π‘Žπ‘žπ‘˜ξ€Έ,ξ€·π‘Ž1,…,π‘Žπ‘šξ€Έπ‘›=ξ€·π‘Ž1,…,π‘Žπ‘šξ€Έ;π‘žπ‘›=ξ€·π‘Ž1ξ€Έ;π‘žπ‘›ξ€·π‘Ž,…,π‘šξ€Έ;π‘žπ‘›.(2.1) Ramanujan’s mock theta function of third-order πœ”(π‘ž) and 𝜈(π‘ž) is

ξ“πœ”(π‘ž)=βˆžπ‘›=0π‘ž2𝑛(𝑛+1)ξ€·π‘ž;π‘ž2ξ€Έ2𝑛+1,(2.2)𝜈(π‘ž)=βˆžπ‘›=0π‘žπ‘›(𝑛+1)ξ€·βˆ’π‘ž;π‘ž2𝑛+1,(2.3)πœ‘(π‘ž)=βˆžπ‘›=βˆ’βˆžπ‘žπ‘›2=ξ€·βˆ’π‘ž;π‘ž2ξ€Έ2βˆžξ€·π‘ž2;π‘ž2ξ€Έβˆž=(βˆ’π‘ž;βˆ’π‘ž)∞(π‘ž;βˆ’π‘ž)∞.(2.4) We will use the following notations for πœƒ-functions.

Definition 2.1. If |π‘ž|<1 and π‘₯β‰ 0, then ξ‚€π‘žπ‘—(π‘₯,π‘ž)=π‘₯,π‘₯,π‘ž;π‘žβˆž.(2.5) If π‘š is a positive integer and π‘Ž is an integer, π½π‘Ž,π‘š=𝑗(π‘žπ‘Ž,π‘žπ‘š),(2.6)π½π‘Ž,π‘š=𝑗(βˆ’π‘žπ‘Ž,π‘žπ‘šπ½),(2.7)π‘šξ€·π‘ž=π‘—π‘š,π‘ž3π‘šξ€Έ=(π‘žπ‘š;π‘žπ‘š)βˆžπ‘—ξ‚€π‘ž,(2.8)π‘₯π‘₯,π‘ž=𝑗(π‘₯,π‘ž),(2.9)𝑗(π‘₯,π‘ž)=βˆ’π‘₯π‘—βˆ’1ξ€Έ,π‘ž,(2.10)𝑗(π‘žπ‘›π‘₯,π‘ž)=(βˆ’1)π‘›π‘žβˆ’π‘›(π‘›βˆ’1)/2π‘₯βˆ’π‘›π‘—(π‘₯,π‘ž),if𝑛isaninteger.(2.11) By Jacobi’s triple product identity [12, page 282] 𝑗(π‘₯,π‘ž)=βˆžπ‘›=βˆ’βˆž(βˆ’1)π‘›π‘žπ‘›(π‘›βˆ’1)/2π‘₯𝑛.(2.12)

2.1. More Definitions

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

ξƒ―πœ€(𝑧)=1if|𝑧|<1,βˆ’1if|𝑧|>1.(2.13)

If s is an integer, then

ξƒ―sg(𝑠)=1if𝑠β‰₯0,βˆ’1if𝑠<0.(2.14)

Using these definitions,

11βˆ’π‘§=πœ€(𝑧)βˆžξ“π‘ =βˆ’βˆžsg(𝑠)=πœ€(𝑧)𝑧𝑠.(2.15) We shall use the following theorems.

Theorem 2.2 (see [13, Theorem 1.3, page 644]). Let π‘ž be fixed, 0<|π‘ž|<1. Let π‘Ž,𝑏, and π‘š be fixed integers with 𝑏≠0 and π‘šβ‰₯1. Define 1𝐹(𝑧)=π‘—ξ€·π‘žπ‘Žπ‘§π‘,π‘žπ‘šξ€Έ.(2.16) Then 𝐹 is meromorphic for 𝑧≠0, with simple poles at all points 𝑧0 such that 𝑧𝑏0=π‘žπ‘˜π‘šβˆ’π‘Ž for some integer π‘˜. The residue of 𝐹(𝑧) at such a point 𝑧0 is (βˆ’1)π‘˜+1π‘žπ‘šπ‘˜(π‘˜βˆ’1)/2𝑧0𝑏𝐽3π‘š.(2.17)

Theorem 2.3 (see [13, Theorem 1.8(a), page 647]). Suppose that 𝐹(𝑧)=π‘ŸπΉπ‘Ÿπ‘§π‘Ÿ(2.18) for all 𝑧≠0 and that 𝐹(𝑧) satisfies 𝐹(π‘žπ‘§)=πΆπ‘§βˆ’π‘›πΉ(𝑧),(2.19) where 0<|π‘ž|<1 and 𝐢≠0. Then 𝐹(𝑧)=π‘›βˆ’1ξ“π‘Ÿ=0πΉπ‘Ÿπ‘§π‘Ÿπ‘—ξ€·βˆ’πΆβˆ’1π‘žπ‘Ÿπ‘§π‘›,π‘žπ‘›ξ€Έ.(2.20) Truesdell [14] calls the functions which satisfy the difference equation πœ•πœ•π‘§πΉ(𝑧,𝛼)=𝐹(𝑧,𝛼+1)(2.21) as 𝐹-function. He unified the study of these 𝐹-functions.
The functions which satisfy the π‘ž-analogue of the difference equation
π·π‘ž,𝑧𝐹(𝑧,𝛼)=𝐹(𝑧,𝛼+1),(2.22) where π‘§π·π‘ž,𝑧𝐹(𝑧,𝛼)=𝐹(𝑧,𝛼)βˆ’πΉ(π‘§π‘ž,𝛼)(2.23) are called πΉπ‘ž-functions.

3. Bilateral π’Ÿ5(π‘ž) as a Sum of Two Mock Theta Functions of Different Orders

(i) We shall denote the bilateral of π’Ÿ5(π‘ž) by π’Ÿ5,𝑐(π‘ž). We define it as

ξ€·π‘ž;π‘ž2ξ€Έ2βˆžπ’Ÿ5,𝑐(π‘ž)=βˆžπ‘›=βˆ’βˆžξ€·π‘ž;π‘ž2ξ€Έ2π‘›π‘ž2𝑛.(3.1) Now

ξ€·π‘ž;π‘ž2ξ€Έ2βˆžπ’Ÿ5,𝑐(π‘ž)=βˆžπ‘›=βˆ’βˆžξ€·π‘ž;π‘ž2ξ€Έ2π‘›π‘ž2𝑛=ξ“βˆžπ‘›=0ξ€·π‘ž;π‘ž2ξ€Έ2π‘›π‘ž2𝑛+ξ“βˆ’βˆžπ‘›=βˆ’1ξ€·π‘ž;π‘ž2ξ€Έ2π‘›π‘ž2𝑛=ξ“βˆžπ‘›=0ξ€·π‘ž;π‘ž2ξ€Έ2π‘›π‘ž2𝑛+ξ“βˆžπ‘›=0π‘ž2𝑛2+2π‘›ξ€·π‘ž;π‘ž2ξ€Έ2𝑛+1,(3.2) and we use (1.2) in the first summation and (2.2) in the second summation, to write

ξ€·π‘ž;π‘ž2ξ€Έ2βˆžπ’Ÿ5,𝑐(π‘ž)=π‘ž;π‘ž2ξ€Έ2βˆžπ’Ÿ5(π‘ž)+πœ”(π‘ž).(3.3)

Thus π’Ÿ5,𝑐(π‘ž) is a sum of a second-order mock theta function and a third-order mock theta function.

(ii) Transformation of Bilateral π’Ÿ5,𝑐(π‘ž) into bilateral πœ”π‘(π‘ž) is as follows.

It is very interesting that the bilateral π’Ÿ5,𝑐(π‘ž) can be written as bilateral third-order mock theta function πœ”π‘(π‘ž).

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

2πœ“2ξ‚Έξ‚Ή=π‘Ž,𝑏𝑐,𝑑;π‘ž,𝑧(π‘Žπ‘§,𝑏𝑧,π‘π‘ž/π‘Žπ‘π‘§,π‘‘π‘ž/π‘Žπ‘π‘§;π‘ž)∞(π‘ž/π‘Ž,π‘ž/𝑏,𝑐,𝑑;π‘ž)βˆžΓ—2πœ“2ξ‚Έξ‚Ήπ‘Žπ‘π‘§/𝑐,π‘Žπ‘π‘§/π‘‘π‘Žπ‘§,𝑏𝑧;π‘ž,𝑐𝑑/π‘Žπ‘π‘§.(3.4) Letting π‘žβ†’π‘ž2, and setting π‘Ž=𝑏=π‘ž, 𝑐=𝑑=0, and 𝑧=π‘ž2 in (3.4), we get

ξ€·π‘ž;π‘ž2ξ€Έ2βˆžπ’Ÿ5,π‘ξ€·π‘ž(π‘ž)=3,π‘ž3;π‘ž2ξ€Έβˆžξ€·π‘ž,π‘ž;π‘ž2ξ€Έβˆžξ“βˆžπ‘›=βˆ’βˆžπ‘ž2𝑛2+2π‘›ξ€·π‘ž3;π‘ž2ξ€Έ2𝑛=ξ“βˆžπ‘›=βˆ’βˆžπ‘ž2𝑛2+2π‘›ξ€·π‘ž;π‘ž2ξ€Έ2𝑛+1=πœ”π‘(π‘ž).(3.5)

4. Another Bilateral Transformation

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

𝑏1,𝑏2,π‘ž/π‘Ž1,π‘ž/π‘Ž2ξ€Έ,𝑑𝑧,π‘ž/𝑑𝑧;π‘žβˆžξ€·π‘1,𝑐2,π‘ž/𝑐1,π‘ž/𝑐2ξ€Έ;π‘žβˆž2πœ“2ξ‚Έπ‘Ž1,π‘Ž2𝑏1,𝑏2ξ‚Ή=π‘ž;π‘ž,𝑧𝑐1𝑐1/π‘Ž1,𝑐1/π‘Ž2,π‘žπ‘1/𝑐1,π‘žπ‘2/𝑐1,𝑑𝑐1𝑧/π‘ž,π‘ž2/𝑑𝑐1𝑧;π‘žβˆžξ€·π‘1,π‘ž/𝑐1,𝑐1/𝑐2,π‘žπ‘2/𝑐1ξ€Έ;π‘žβˆž2πœ“2ξ‚Έπ‘žπ‘Ž1/𝑐1,π‘žπ‘Ž2/𝑐1π‘žπ‘1/𝑐1,π‘žπ‘2/𝑐1𝑐;π‘ž,𝑧+idem1;𝑐2ξ€Έ,(4.1) where 𝑑=π‘Ž1π‘Ž2/𝑐1𝑐2, |𝑏1𝑏2/π‘Ž1π‘Ž2|<|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 2πœ“2 series on the left side of (4.1). This gives us the freedom to choose the c’s in a convenient way.

Letting π‘žβ†’π‘ž2 and setting, π‘Ž1=π‘Ž2=π‘ž, 𝑏1=𝑏2=0, and 𝑧=π‘ž2 in (4.1), so 𝑑=π‘ž2/𝑐1𝑐2 and 0<|𝑧|<1, to get

ξ€·π‘ž;π‘ž2ξ€Έ4βˆžξ€·π‘ž4/𝑐1𝑐2;π‘ž2ξ€Έβˆžξ€·π‘1𝑐2/π‘ž2;π‘ž2ξ€Έβˆžξ€·π‘2;π‘ž2ξ€Έβˆžξ€·π‘ž2/𝑐2;π‘ž2ξ€Έβˆžπ’Ÿ5,𝑐=π‘ž(π‘ž)2𝑐1𝑐1/π‘ž;π‘ž2ξ€Έ2βˆžξ€·π‘ž2/𝑐2;π‘ž2ξ€Έβˆžξ€·π‘2;π‘ž2ξ€Έβˆžξ€·π‘1/𝑐2;π‘ž2ξ€Έβˆžξ€·π‘ž2𝑐2/𝑐1;π‘ž2ξ€Έβˆžξ“βˆžπ‘›=βˆ’βˆžξ€·π‘ž3/𝑐1;π‘ž2ξ€Έ2π‘›π‘ž2𝑛𝑐+idem1;𝑐2ξ€Έ.(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 π’Ÿ5,𝑐(π‘ž)

We define the bilateral generalized function π’Ÿ5,𝑐(𝑧,𝛼) as

ξ€·π‘ž;π‘ž2ξ€Έ2βˆžπ’Ÿ5,𝑐1(𝑧,𝛼)=(𝑧)βˆžξ“βˆžπ‘›=βˆ’βˆžξ€·π‘ž;π‘ž2ξ€Έ2𝑛(𝑧)π‘›π‘žπ‘›π›Ό+𝑛.(5.1) For 𝛼=1, 𝑧=0, π’Ÿ5,𝑐(𝑧,𝛼) reduce to π’Ÿ5,𝑐(π‘ž).

Now

π·π‘ž,π‘§ξ€Ίπ’Ÿ5,𝑐=1(𝑧,𝛼)π‘§ξ€Ίπ’Ÿ5,𝑐(𝑧,𝛼)βˆ’π’Ÿ5,𝑐=1(π‘§π‘ž,𝛼)π‘§ξ€·π‘ž;π‘ž2ξ€Έ2βˆžξ‚Έ1(𝑧)βˆžξ“βˆžπ‘›=βˆ’βˆžξ€·π‘ž;π‘ž2ξ€Έ2𝑛(𝑧)π‘›π‘žπ‘›π›Ό+π‘›βˆ’1(π‘§π‘ž)βˆžξ“βˆžπ‘›=βˆ’βˆžξ€·π‘ž;π‘ž2ξ€Έ2𝑛(π‘§π‘ž)π‘›π‘žπ‘›π›Ό+𝑛=1π‘§ξ€·π‘ž;π‘ž2ξ€Έ2∞1(𝑧)βˆžξ“βˆžπ‘›=βˆ’βˆžξ€·π‘ž;π‘ž2ξ€Έ2𝑛(𝑧)π‘›π‘žπ‘›π›Ό+𝑛(1βˆ’(1βˆ’π‘§π‘žπ‘›=1))ξ€·π‘ž;π‘ž2ξ€Έ2∞(𝑧)βˆžξ“βˆžπ‘›=βˆ’βˆžξ€·π‘ž;π‘ž2ξ€Έ2𝑛(𝑧)π‘›π‘žπ‘›π›Ό+2𝑛=π’Ÿ5,𝑐(𝑧,𝛼+1).(5.2) So π’Ÿ5,𝑐(𝑧,𝛼+1) is an πΉπ‘ž-function.

Being πΉπ‘ž-function it has unified properties of πΉπ‘ž-functions. For example, one has the following.

(i) The inverse operator π·βˆ’1π‘ž,π‘₯ of π‘ž-differentiation is related to π‘ž-integration as

π·βˆ’1π‘ž,π‘₯πœ™(π‘₯)=(1βˆ’π‘ž)βˆ’1ξ€œπœ™(π‘₯)π‘‘π‘žπ‘₯.(5.3) See Jackson [16].

(ii) π·π‘›π‘ž,π‘§πΉπ‘ž(𝑧,𝛼)=πΉπ‘ž(𝑧,𝛼+𝑛), where 𝑛 is a nonnegative integer.

6. Behaviour of π’Ÿ5(π‘ž) outside the Unit Circle

By definition (1.1)

π’Ÿ5(π‘ž)=βˆžπ‘›=0(βˆ’π‘ž;π‘ž)π‘›ξ€·π‘ž;π‘ž2𝑛+1π‘žπ‘›.(6.1) Replacing π‘ž by 1/π‘ž and writing π’Ÿβˆ—5(π‘ž) for π’Ÿ5(1/π‘ž) [10],

π’Ÿβˆ—5(π‘ž)=βˆžπ‘›=0(βˆ’1)π‘›π‘ž(𝑛2+𝑛)/2(βˆ’π‘ž;π‘ž)π‘›ξ€·π‘ž;π‘ž2𝑛+1=1βˆ’π‘ž2+π‘ž6βˆ’π‘ž12+π‘ž20βˆ’π‘ž30=+β‹―βˆžπ‘›=0(βˆ’1)π‘›π‘žπ‘›2+𝑛,(6.2) which is a πœƒ-function.

7. Lambert Series Expansion for β„Ž1(π‘ž)

For the double series expansion, we first require the generalized Lambert series expansion for β„Ž1(π‘ž).

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

π’Ÿ5(π‘ž)=2β„Ž1(π‘ž)βˆ’(βˆ’π‘ž;π‘ž)2βˆžπœ”(π‘ž),(7.1) where

β„Ž1(π‘ž)=βˆžπ‘›=0(βˆ’π‘ž;π‘ž)2π‘›ξ€·π‘ž;π‘ž2ξ€Έ2𝑛+1π‘žπ‘›.(7.2) There is a slight misprint in the definition β„Ž1(π‘ž) 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 β„Ž1(π‘ž) is

β„Ž1(π‘ž)=βˆžπ‘›=0(βˆ’π‘ž;π‘ž)2π‘›ξ€·π‘ž;π‘ž2ξ€Έ2𝑛+1π‘žπ‘›=1πœƒ4(0,π‘ž)βˆžπ‘›=0(βˆ’1)π‘›π‘žπ‘›(𝑛+2)1βˆ’π‘ž2𝑛+1=12πœƒ4(0,π‘ž)βˆžπ‘›=βˆ’βˆž(βˆ’1)π‘›π‘žπ‘›(𝑛+2)1βˆ’π‘ž2𝑛+1.(7.3)

8. β„Ž1(π‘ž) as a Coefficient of 𝑧0 of a πœƒ-Function

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

ξ“βˆžπ‘Ÿ=βˆ’βˆžπ‘₯π‘Ÿ1βˆ’π‘žπ‘Ÿπ‘¦=𝐽31𝑗(π‘₯𝑦,π‘ž)𝑗(π‘₯,π‘ž)𝑗(𝑦,π‘ž)(8.1) let π‘žβ†’π‘ž2, and then put 𝑦=π‘ž, to get

ξ“βˆžπ‘Ÿ=βˆ’βˆžπ‘₯π‘Ÿ1βˆ’π‘ž2π‘Ÿ+1=𝐽32π‘—ξ€·π‘žπ‘₯,π‘ž2𝑗π‘₯,π‘ž2ξ€Έπ‘—ξ€·π‘ž,π‘ž2ξ€Έ.(8.2) For |π‘ž|<1, and 𝑧≠0 and not an integral power of π‘ž, let

𝐴1(𝑧)=2πœƒ4𝐽(0,π‘ž)32π‘—ξ€·π‘žπ‘§,π‘ž2𝑗𝑧,π‘ž2ξ€Έπ‘—ξ€·π‘ž,π‘ž2ξ€Έπ‘—ξ‚΅π‘§π‘ž,π‘ž2ξ‚Ά.(8.3)

Theorem 8.1. Let π‘ž be fixed with 0<|π‘ž|<1. Then β„Ž1(π‘ž) is the coefficient of 𝑧0 in the Laurent series expansion of 𝐴(𝑧) in the annulus |π‘ž|<|𝑧|<1.

Proof. By (7.3) 2πœƒ4(0,π‘ž)β„Ž1(π‘ž)=βˆžπ‘›=βˆ’βˆž(βˆ’1)π‘›π‘žπ‘›(𝑛+2)1βˆ’π‘ž2𝑛+1=coefficientof𝑧0inβˆžπ‘›=βˆ’βˆžπ‘§π‘›1βˆ’π‘ž2𝑛+1ξ“βˆžπ‘ =βˆ’βˆž(βˆ’1)π‘ π‘žπ‘ 2+π‘ ξ‚΅π‘§π‘žξ‚Άβˆ’π‘ =coefficientof𝑧0𝐽in32π‘—ξ€·π‘žπ‘§,π‘ž2𝑗𝑧,π‘ž2ξ€Έπ‘—ξ€·π‘ž,π‘ž2ξ€Έπ‘—ξ‚΅π‘§π‘ž,π‘ž2ξ‚Ά(8.4) dividing by 2πœƒ4(0,π‘ž) gives the theorem.

9. An Identity for β„Ž1(π‘ž)

Theorem 9.1. If 0<|π‘ž|<1 and z is neither zero nor an integral power of π‘ž, then 𝐴(𝑧,π‘ž)=𝑗𝑧,π‘ž2ξ€Έβ„Ž1βˆ’ξ‚€βˆ’1(π‘ž)2ξ‚ξ“βˆžπ‘Ÿ=βˆ’βˆž(βˆ’1)π‘Ÿπ‘žπ‘Ÿ2+3π‘Ÿβˆ’1π‘§π‘Ÿ+21βˆ’π‘ž2π‘Ÿ+2π‘§βˆ’12ξ“βˆžπ‘Ÿ=βˆ’βˆž(βˆ’1)π‘Ÿπ‘žπ‘Ÿ2+3π‘Ÿ+1π‘§βˆ’π‘Ÿβˆ’11βˆ’π‘ž2π‘Ÿ+2π‘§βˆ’1.(9.1) Define 1𝐿(𝑧)=βˆ’2ξ“βˆžπ‘Ÿ=βˆ’βˆž(βˆ’1)π‘Ÿπ‘žπ‘Ÿ2+3π‘Ÿβˆ’1π‘§π‘Ÿ+21βˆ’π‘ž2π‘Ÿ+2𝑧1(9.2)𝑀(𝑧)=2ξ“βˆžπ‘Ÿ=βˆ’βˆž(βˆ’1)π‘Ÿπ‘žπ‘Ÿ2+3π‘Ÿ+1π‘§βˆ’π‘Ÿβˆ’11βˆ’π‘ž2π‘Ÿ+2π‘§βˆ’1𝐹,(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: πΉξ€·π‘ž2𝑧=βˆ’π‘§βˆ’1𝐹(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 πΉξ€·π‘ž2𝑧=βˆ’π‘§βˆ’1𝐹(𝑧).(9.6) We shall show that each of 𝐴(𝑧), 𝐿(𝑧), and 𝑀(𝑧) satisfies the functional equation: 𝐴1(𝑧)=2(βˆ’π‘ž;π‘ž)∞(π‘ž;π‘ž)∞𝐽32π‘—ξ€·π‘žπ‘§,π‘ž2𝑗𝑧,π‘ž2ξ€Έπ‘—ξ€·π‘ž,π‘ž2ξ€Έπ‘—ξ‚΅π‘§π‘ž,π‘ž2ξ‚Ά,(9.7) and so π΄ξ€·π‘ž2𝑧=12(βˆ’π‘ž;π‘ž)∞(π‘ž;π‘ž)∞𝐽32π‘—ξ€·π‘ž3𝑧,π‘ž2ξ€Έπ‘—ξ€·π‘ž2𝑧,π‘ž2ξ€Έπ‘—ξ€·π‘ž,π‘ž2ξ€Έπ‘—ξ€·π‘§π‘ž,π‘ž2ξ€Έ.(9.8) We employ (2.11) on the right-hand side to get π΄ξ€·π‘ž2𝑧=12(βˆ’π‘ž;π‘ž)∞(π‘ž;π‘ž)∞𝐽32(βˆ’1)π‘§βˆ’1π‘žβˆ’1π‘—ξ€·π‘§π‘ž,π‘ž2ξ€Έ(βˆ’1)π‘žπ‘§βˆ’1(βˆ’1)π‘§βˆ’1𝑗𝑧,π‘ž2ξ€Έπ‘—ξ€·π‘ž,π‘ž2ξ€Έπ‘—ξ‚΅π‘§π‘ž,π‘ž2ξ‚Άπ΄ξ€·π‘ž2𝑧=βˆ’π‘§βˆ’1𝐴(𝑧).(9.9) We now take 𝐿(𝑧): πΏξ€·π‘ž2𝑧=12ξ“βˆžπ‘Ÿ=βˆ’βˆž(βˆ’1)π‘Ÿπ‘žπ‘Ÿ2+3π‘Ÿβˆ’1ξ€·π‘§π‘ž2ξ€Έπ‘Ÿ+21βˆ’π‘ž2π‘Ÿ+2ξ€·π‘§π‘ž2ξ€Έ.(9.10) Writing π‘Ÿβˆ’1 for π‘Ÿon the right-hand side we have πΏξ€·π‘ž2𝑧=βˆ’π‘§βˆ’1𝐿(𝑧).(9.11) Similarly only writing π‘Ÿ+1 for π‘Ÿ we have π‘€ξ€·π‘ž2𝑧=βˆ’π‘§βˆ’1𝑀(𝑧).(9.12) Hence the functional equation (9.4) is proved.
Obviously 𝐿(𝑧) and 𝑀(𝑧) are meromorphic for 𝑧≠0. 𝐿(𝑧) has simple poles at 𝑧=π‘ž2π‘˜βˆ’2 and 𝑀(𝑧) has simple poles at 𝑧=π‘ž2π‘˜+2. Hence 𝐹(𝑧) is meromorphic for 𝑧≠0 with, at most, simple poles at 𝑧=π‘ž2π‘˜Β±2.
Taking π‘Ÿ=0 in (9.2), we calculate the residue of 𝐿(𝑧) at the point 𝑧=1/π‘ž2:
Residueof𝐿(𝑧)=lim𝑧→1/π‘ž212ξ‚΅1π‘§βˆ’π‘ž2𝑧2π‘žβˆ’1ξ€·ξ€·π‘§βˆ’1/π‘ž2π‘žξ€Έξ€Έ2=12π‘žβˆ’5.(9.13) For the residue of 𝐴(𝑧) at 𝑧=1/π‘ž2, take 𝑏=1, π‘˜=βˆ’1, π‘š=2, π‘Ž=0 in (2.16) to get 1Residueof𝐴(𝑧)=2(βˆ’π‘ž;π‘ž)∞(π‘ž;π‘ž)∞𝐽32𝑗1/π‘ž,π‘ž2𝑗1/π‘ž3,π‘ž2ξ€Έπ‘—ξ€·π‘ž,π‘ž2ξ€Έ1𝐽32=12(βˆ’π‘ž;π‘ž)∞(π‘ž;π‘ž)βˆžπ‘—ξ€·π‘ž,π‘ž2ξ€Έπ‘—ξ€·π‘ž3,π‘ž2ξ€Έπ‘ž4π‘—ξ€·π‘ž,π‘ž2ξ€Έ=12(βˆ’π‘ž;π‘ž)∞(π‘ž;π‘ž)∞1π‘ž4ξ€·π‘ž3;π‘ž2ξ€Έβˆžξ‚΅1π‘ž;π‘ž2ξ‚Άβˆžξ€·π‘ž2;π‘ž2ξ€Έβˆž=12(βˆ’π‘ž;π‘ž)∞(π‘ž;π‘ž)∞1π‘ž4(1βˆ’(1/π‘ž))ξ€·(1βˆ’π‘ž)π‘ž;π‘ž2ξ€Έβˆžξ€·π‘ž;π‘ž2ξ€Έβˆžξ€·π‘ž2;π‘ž2ξ€Έβˆž1=βˆ’2(βˆ’π‘ž;π‘ž)∞(π‘ž;π‘ž)∞1π‘ž5(π‘ž;π‘ž)∞(βˆ’π‘ž;π‘ž)∞1=βˆ’2π‘žβˆ’5.(9.14) So the residue of 𝐹(𝑧) at 𝑧=1/π‘ž2 is βˆ’(1/2)π‘ž5+0+(1/2)π‘ž5=0.
Now we calculate the residue at 𝑧=π‘ž2:
Residueof𝑀(𝑧)=limπ‘§β†’π‘ž212ξ€·π‘§βˆ’π‘ž2ξ€Έπ‘žπ‘§βˆ’1ξ€·1βˆ’π‘ž2π‘§βˆ’1ξ€Έ=limπ‘§β†’π‘ž212ξ€·π‘§βˆ’π‘ž2ξ€Έπ‘žξ€·π‘§βˆ’π‘ž2ξ€Έ=π‘ž2,(9.15) and for the residue of 𝐴(𝑧) at 𝑧=π‘ž2, taking 𝑏=1, π‘˜=1, π‘š=2, and π‘Ž=0 in (2.16), so 1Residueof𝐴(𝑧)=2(βˆ’π‘ž;π‘ž)∞(π‘ž;π‘ž)∞𝐽32π‘—ξ€·π‘ž3,π‘ž2ξ€Έπ‘—ξ€·π‘ž,π‘ž2ξ€Έπ‘—ξ€·π‘ž,π‘ž2ξ€Έπ‘ž2𝐽32=12(βˆ’π‘ž;π‘ž)∞(π‘ž;π‘ž)βˆžπ‘—ξ€·π‘ž3,π‘ž2ξ€Έπ‘ž2=12(βˆ’π‘ž;π‘ž)∞(π‘ž;π‘ž)βˆžξ€·π‘ž3;π‘ž2ξ€Έβˆžξ‚΅1π‘ž;π‘ž2ξ‚Άβˆžξ€·π‘ž2;π‘ž2ξ€Έβˆžπ‘ž2=12(βˆ’π‘ž;π‘ž)∞(π‘ž;π‘ž)∞(1βˆ’(1/π‘ž))ξ€·(1βˆ’π‘ž)π‘ž;π‘ž2ξ€Έβˆžξ€·π‘ž;π‘ž2ξ€Έβˆžξ€·π‘ž2;π‘ž2ξ€Έβˆžπ‘ž21=βˆ’2π‘ž.(9.16) Hence the residue of 𝐹(𝑧) at 𝑧=π‘ž2 is 0+(1/2)π‘žβˆ’(1/2)π‘ž=0. Hence 𝐹(𝑧) is analytic at 𝑧=π‘ž2.
Since 𝐹(𝑧) satisfies (9.4), so 𝐹(𝑧) is analytic at all points of the form 𝑧=π‘ž2π‘˜Β±2 and hence for all 𝑧≠0.
We now apply (2.20) with 𝑛=1 and 𝑐=βˆ’1 and π‘ž replaced by π‘ž2 to get
𝐹(𝑧)=𝐹0𝑗𝑧,π‘ž2ξ€Έ,(9.17) where 𝐹0 is the coefficient of 𝑧0 in the Laurent expansion of 𝐹(𝑧),𝑧≠0.
Now for |π‘ž|<|𝑧|<1, by Theorem 8.1, the coefficient of 𝑧0 in 𝐴(𝑧) is β„Ž1(π‘ž).
For such 𝑧, |π‘ž2π‘Ÿ+2𝑧|<1 if and only if π‘Ÿβ‰₯0.
That is,
πœ€ξ€·π‘ž2π‘Ÿ+2𝑧=sg(π‘Ÿ).(9.18) Hence by (2.15) 11βˆ’π‘ž2π‘Ÿ+2𝑧=sg(π‘Ÿ)βˆžξ“π‘Ÿ=βˆ’βˆžsg(π‘Ÿ)=sg(𝑠)π‘ž(2π‘Ÿ+2)𝑠𝑧𝑠.(9.19) So 1𝐿(𝑧)=βˆ’2ξ“βˆžsg(π‘Ÿ)=sg(𝑠)sg(π‘Ÿ)(βˆ’1)π‘Ÿπ‘žπ‘Ÿ2+3π‘Ÿβˆ’1+(2π‘Ÿ+2)π‘ π‘§π‘Ÿ+2+𝑠.(9.20) If sg(π‘Ÿ)=sg(𝑠), then π‘Ÿ+𝑠+2 is either β‰₯1 or β‰€βˆ’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 β„Ž1(π‘ž).
Hence by (9.17), we have
𝐹(𝑧)=β„Ž1ξ€·(π‘ž)𝑗𝑧,π‘ž2ξ€Έ,(9.21) which gives the theorem.

10. Double Series Expansion

Now we derive the double series expansion for β„Ž1(π‘ž). 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}, 𝑛β‰₯0, form a Bailey pair relative to a number π‘Ž if

𝛽𝑛=ξ“π‘›π‘Ÿ=0π›Όπ‘Ÿ(π‘ž)π‘›βˆ’π‘Ÿ(π‘Žπ‘ž)𝑛+π‘Ÿ,(10.1) for all 𝑛β‰₯0.

Corollary 10.1 (see [5, Corollary. 2.1, page 70]). If {𝛼𝑛} and {𝛽𝑛} form a Bailey pair relative to π‘Ž, then ξ“βˆžπ‘›=0ξ€·πœŒ1ξ€Έπ‘›ξ€·πœŒ2ξ€Έπ‘›ξ€·π‘Žπ‘ž/𝜌1𝜌2ξ€Έπ‘›π›Όπ‘›ξ€·π‘Žπ‘ž/𝜌1ξ€Έπ‘›ξ€·π‘Žπ‘ž/𝜌2𝑛=(π‘Žπ‘ž)βˆžξ€·π‘Žπ‘ž/𝜌1𝜌2ξ€Έβˆžξ€·π‘Žπ‘ž/𝜌1ξ€Έβˆžξ€·π‘Žπ‘ž/𝜌2ξ€Έβˆžξ“βˆžπ‘›=0ξ€·πœŒ1ξ€Έπ‘›ξ€·πœŒ2ξ€Έπ‘›ξ‚΅π‘Žπ‘žπœŒ1𝜌2𝑛𝛽𝑛(10.2) provided that both sums converge absolutely.

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

Let π‘Ž, 𝑏, 𝑐, and π‘ž be complex numbers with π‘Žβ‰ 1, 𝑏≠0, 𝑐≠0, π‘žβ‰ 0, and none π‘Ž/𝑏, π‘Ž/𝑐, π‘žπ‘, π‘žπ‘ of the form π‘žβˆ’π‘˜ with π‘˜β‰₯0. For 𝑛β‰₯0, define

π΄ξ…žπ‘›π‘ž(π‘Ž,𝑏,𝑐,π‘ž)=𝑛2(𝑏𝑐)𝑛1βˆ’π‘Žπ‘ž2𝑛(π‘Ž/𝑏)𝑛(π‘Ž/𝑐)𝑛(1βˆ’π‘Ž)(π‘žπ‘)𝑛(π‘žπ‘)𝑛×𝑛𝑗=0(βˆ’1)𝑗1βˆ’π‘Žπ‘ž2π‘—βˆ’1ξ€Έ(π‘Ž)π‘—βˆ’1(𝑏)𝑗(𝑐)π‘—π‘ž(𝑗2)(𝑏𝑐)𝑗(π‘ž)𝑗(π‘Ž/𝑏)𝑗(π‘Ž/𝑐)𝑗,π΅ξ…žπ‘›1(π‘Ž,𝑏,𝑐,π‘ž)=(π‘žπ‘)𝑛(π‘žπ‘)𝑛.(10.3) Then the sequences {π΄ξ…žπ‘›(π‘Ž,𝑏,𝑐,π‘ž)} and {π΅ξ…žπ‘›(π‘Ž,𝑏,𝑐,π‘ž)} form a Bailey pair relative to π‘Ž.

Letting π‘žβ†’π‘ž2 and then taking π‘Ž=π‘ž2, 𝑏=𝑐=π‘ž, in (10.3), we get

π΄ξ…žπ‘›ξ€·π‘ž2,π‘ž,π‘ž2ξ€Έ=π‘ž2𝑛2+2𝑛1βˆ’π‘ž4𝑛+2ξ€Έξ€·π‘ž;π‘ž2ξ€Έ2𝑛1βˆ’π‘ž2π‘žξ€Έξ€·3;π‘ž2ξ€Έ2𝑛×𝑛𝑗=0(βˆ’1)𝑗1βˆ’π‘ž4π‘—π‘žξ€Έξ€·2;π‘ž2ξ€Έπ‘—βˆ’1ξ€·π‘ž;π‘ž2ξ€Έ2π‘—π‘žπ‘—2+π‘—ξ€·π‘ž2;π‘ž2ξ€Έπ‘—ξ€·π‘ž;π‘ž2ξ€Έ2𝑗=(1βˆ’π‘ž)2ξ€·1+π‘ž2𝑛+1ξ€Έπ‘ž2𝑛2+2𝑛1βˆ’π‘ž2ξ€Έξ€·1βˆ’π‘ž2𝑛+1𝑛𝑗=0(βˆ’1)π‘—π‘žβˆ’π‘—2βˆ’π‘—ξ€·1+π‘ž2𝑗=(1βˆ’π‘ž)2ξ€·1+π‘ž2𝑛+1ξ€Έπ‘ž2𝑛2+2𝑛1βˆ’π‘ž2ξ€Έξ€·1βˆ’π‘ž2𝑛+11+𝑛𝑗=βˆ’π‘›(βˆ’1)π‘—π‘žβˆ’π‘—2βˆ’π‘—ξ‚„,π΅ξ…žπ‘›ξ€·π‘ž2,π‘ž,π‘ž,π‘ž2ξ€Έ=1ξ€·π‘ž3;π‘ž2ξ€Έ2𝑛.(10.4) Now letting π‘žβ†’π‘ž2 and then setting 𝜌1=βˆ’π‘ž, 𝜌2=βˆ’π‘ž2, π‘Ž=π‘ž2 in (10.2) we get

ξ“βˆžπ‘›=0π‘žπ‘›ξ€·βˆ’π‘ž;π‘ž2ξ€Έπ‘›ξ€·βˆ’π‘ž3;π‘ž2𝑛𝛼𝑛=ξ€·π‘ž;π‘ž2ξ€Έβˆžξ€·π‘ž4;π‘ž2ξ€Έβˆžξ€·βˆ’π‘ž2;π‘ž2ξ€Έβˆžξ€·βˆ’π‘ž3;π‘ž2ξ€Έβˆžξ“βˆžπ‘›=0ξ€·βˆ’π‘ž;π‘ž2ξ€Έπ‘›ξ€·βˆ’π‘ž2;π‘ž2ξ€Έπ‘›π‘žπ‘›π›½π‘›.(10.5) Taking π΄ξ…žπ‘› and π΅ξ…žπ‘› for π›Όξ…žπ‘› and π›½ξ…žπ‘›, respectively, in (10.5) and using the definition of β„Ž1(π‘ž), we get

ξ€·π‘ž;π‘ž2ξ€Έβˆžξ€·π‘ž2;π‘ž2ξ€Έβˆžξ€·βˆ’π‘ž;π‘ž2ξ€Έβˆžξ€·βˆ’π‘ž2;π‘ž2ξ€Έβˆžβ„Ž1(π‘ž)=βˆžπ‘›=0π‘ž2𝑛2+3𝑛1βˆ’π‘ž2𝑛+11+𝑛𝑗=βˆ’π‘›(βˆ’1)π‘—π‘žβˆ’π‘—2βˆ’π‘—ξ‚„(10.6) or

β„Ž1ξ€·(π‘ž)=βˆ’π‘ž;π‘ž2ξ€Έβˆžξ€·βˆ’π‘ž2;π‘ž2ξ€Έβˆžξ€·π‘ž;π‘ž2ξ€Έβˆžξ€·π‘ž2;π‘ž2ξ€Έβˆžξ“βˆžπ‘›=0π‘ž2𝑛2+3𝑛1βˆ’π‘ž2𝑛+11+𝑛𝑗=βˆ’π‘›(βˆ’1)π‘—π‘žβˆ’π‘—2βˆ’π‘—ξ‚„,(10.7) which is the double series expansion for β„Ž1(π‘ž).

This double series expansion can be used to get more properties of π’Ÿ5(π‘ž).

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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.


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