Abstract

We consider the second-order mock theta function 𝒟5 (ğ‘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 𝒟5 (ğ‘ž) outside the unit circle is a theta function and also write ℎ1(ğ‘ž) as a coefficient of 𝑧0 of a theta series. First writing ℎ1(ğ‘ž) as a coefficient of a theta function, we prove an identity for ℎ1(ğ‘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 “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(î“ğ‘ž)=âˆžğ‘›=0ğ‘žğ‘›(âˆ’ğ‘ž;ğ‘ž)ğ‘›î€·ğ‘ž;ğ‘ž2𝑛+1=1(1.1)(ğ‘ž;ğ‘ž2)2âˆžî“âˆžğ‘›=0î€·ğ‘ž;ğ‘ž22ğ‘›ğ‘ž2𝑛(1.2) and proved that 𝒟5(ğ‘ž) is a mock theta function and called it of “2nd” order.

He further showed that 𝒟5(ğ‘ž) is a sum of two mock theta functions ℎ1(ğ‘ž) and 𝜔(ğ‘ž) where ℎ1(ğ‘ž) is of second-order and 𝜔(ğ‘ž) is Ramanujan’s mock theta function of third-order. This 𝒟5(ğ‘ž) will be the basis of our study in this paper.

Before we begin with the study of 𝒟5(ğ‘ž) and ℎ1(ğ‘ž) it will be appropriate to mention the work done earlier.

Gordon and McIntosh in their survey paper [8] have shown that ℎ1(ğ‘ž) is essentially the odd part of the second-order mock theta function 𝐵(ğ‘ž), which appears as 𝛽(ğ‘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,

ℎ1î€·ğ‘ž2=𝐵(ğ‘ž)−𝐵(âˆ’ğ‘ž)4ğ‘ž,(1.3) where

𝐵(ğ‘ž)=âˆžğ‘›=0ğ‘žğ‘›(𝑛+1)î€·âˆ’ğ‘ž2;ğ‘ž2î€¸ğ‘›î€·ğ‘ž;ğ‘ž22𝑛+1=î“âˆžğ‘›=0ğ‘žğ‘›î€·âˆ’ğ‘ž;ğ‘ž2î€¸ğ‘›î€·ğ‘ž;ğ‘ž2𝑛+1.(1.4) Since the even part of 𝐵(ğ‘ž) is the ordinary theta function

𝐵(ğ‘ž)+𝐵(âˆ’ğ‘ž)2=î€·ğ‘ž4;ğ‘ž4î€¸âˆžî€·âˆ’ğ‘ž2;ğ‘ž24∞,(1.5) it follows that the odd part and ℎ1(ğ‘ž) are second-order mock theta functions. Thus 𝒟5(ğ‘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 𝒟5(ğ‘ž) as a bilateral q-series and show that it is also a sum of the second-order mock theta function 𝒟5(ğ‘ž) and the third-order mock theta function 𝜔(ğ‘ž). By using Bailey’s transformation we have the interesting result that the bilateral 𝒟5,𝑐(ğ‘ž) is the same as the bilateral 𝜔𝑐(ğ‘ž).

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

In Section 5, a mild generalization 𝒟5,𝑐(𝑧,𝛼) of 𝒟5,𝑐(ğ‘ž) is given and we show that this generalized function is a ğ¹ğ‘ž-function.

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

In Section 7 we state a generalized Lambert Series expansion for ℎ1(ğ‘ž) as given in [8].

In Section 8 we show that ℎ1(ğ‘ž) is a coefficient of 𝑧0 of a theta function.

In Section 9 we prove an identity for ℎ1(ğ‘ž) using ℎ1(ğ‘ž) as a coefficient of 𝑧0 of a theta function.

In Section 10 a double series expansion for ℎ1(ğ‘ž) is obtained by using Bailey pair method.

2. Basic Preliminaries

We first introduce some standard notation.

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

(ğ‘Ž)0=(ğ‘Ž;ğ‘ž)0=1,(ğ‘Ž)𝑛=(ğ‘Ž;ğ‘ž)𝑛=𝑛−1𝑘=01âˆ’ğ‘Žğ‘žğ‘˜î€¸,(ğ‘Ž)∞=(ğ‘Ž;ğ‘ž)∞=âˆžî‘ğ‘˜=01âˆ’ğ‘Žğ‘žğ‘˜î€¸,î€·ğ‘Ž1,…,ğ‘Žğ‘šî€¸ğ‘›=î€·ğ‘Ž1,…,ğ‘Žğ‘šî€¸;ğ‘žğ‘›=î€·ğ‘Ž1;ğ‘žğ‘›î€·ğ‘Ž,…,𝑚;ğ‘žğ‘›.(2.1) Ramanujan’s mock theta function of third-order 𝜔(ğ‘ž) and 𝜈(ğ‘ž) is

𝜔(ğ‘ž)=âˆžğ‘›=0ğ‘ž2𝑛(𝑛+1)î€·ğ‘ž;ğ‘ž22𝑛+1,(2.2)𝜈(ğ‘ž)=âˆžğ‘›=0ğ‘žğ‘›(𝑛+1)î€·âˆ’ğ‘ž;ğ‘ž2𝑛+1,(2.3)𝜑(ğ‘ž)=âˆžğ‘›=âˆ’âˆžğ‘žğ‘›2=î€·âˆ’ğ‘ž;ğ‘ž22âˆžî€·ğ‘ž2;ğ‘ž2∞=(âˆ’ğ‘ž;âˆ’ğ‘ž)∞(ğ‘ž;âˆ’ğ‘ž)∞.(2.4) We will use the following notations for 𝜃-functions.

Definition 2.1. If |ğ‘ž|<1 and 𝑥≠0, then î‚€ğ‘žğ‘—(𝑥,ğ‘ž)=𝑥,𝑥,ğ‘ž;ğ‘žâˆž.(2.5) If 𝑚 is a positive integer and ğ‘Ž is an integer, ğ½ğ‘Ž,𝑚=𝑗(ğ‘žğ‘Ž,ğ‘žğ‘š),(2.6)ğ½ğ‘Ž,𝑚=𝑗(âˆ’ğ‘žğ‘Ž,ğ‘žğ‘šğ½),(2.7)ğ‘šî€·ğ‘ž=𝑗𝑚,ğ‘ž3𝑚=(ğ‘žğ‘š;ğ‘žğ‘š)âˆžğ‘—î‚€ğ‘ž,(2.8)𝑥𝑥,ğ‘ž=𝑗(𝑥,ğ‘ž),(2.9)𝑗(𝑥,ğ‘ž)=−𝑥𝑗−1,ğ‘ž,(2.10)𝑗(ğ‘žğ‘›ğ‘¥,ğ‘ž)=(−1)ğ‘›ğ‘žâˆ’ğ‘›(𝑛−1)/2𝑥−𝑛𝑗(𝑥,ğ‘ž),if𝑛isaninteger.(2.11) By Jacobi’s triple product identity [12, page 282] 𝑗(𝑥,ğ‘ž)=âˆžğ‘›=−∞(−1)ğ‘›ğ‘žğ‘›(𝑛−1)/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)=1if𝑠≥0,−1if𝑠<0.(2.14)

Using these definitions,

11−𝑧=𝜀(𝑧)âˆžî“ğ‘ =−∞sg(𝑠)=𝜀(𝑧)𝑧𝑠.(2.15) We shall use the following theorems.

Theorem 2.2 (see [13, Theorem 1.3, page 644]). Let ğ‘ž be fixed, 0<|ğ‘ž|<1. Let ğ‘Ž,𝑏, and 𝑚 be fixed integers with 𝑏≠0 and 𝑚≥1. Define 1𝐹(𝑧)=ğ‘—î€·ğ‘žğ‘Žğ‘§ğ‘,ğ‘žğ‘šî€¸.(2.16) Then 𝐹 is meromorphic for 𝑧≠0, with simple poles at all points 𝑧0 such that 𝑧𝑏0=ğ‘žğ‘˜ğ‘šâˆ’ğ‘Ž for some integer 𝑘. The residue of 𝐹(𝑧) at such a point 𝑧0 is (−1)𝑘+1ğ‘žğ‘šğ‘˜(𝑘−1)/2𝑧0𝑏𝐽3𝑚.(2.17)

Theorem 2.3 (see [13, Theorem 1.8(a), page 647]). Suppose that 𝐹(𝑧)=𝑟𝐹𝑟𝑧𝑟(2.18) for all 𝑧≠0 and that 𝐹(𝑧) satisfies 𝐹(ğ‘žğ‘§)=𝐶𝑧−𝑛𝐹(𝑧),(2.19) where 0<|ğ‘ž|<1 and 𝐶≠0. Then 𝐹(𝑧)=𝑛−1𝑟=0𝐹𝑟𝑧𝑟𝑗−𝐶−1ğ‘žğ‘Ÿğ‘§ğ‘›,ğ‘žğ‘›î€¸.(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 ğ‘ž-analogue of the difference equation
ğ·ğ‘ž,𝑧𝐹(𝑧,𝛼)=𝐹(𝑧,𝛼+1),(2.22) where ğ‘§ğ·ğ‘ž,𝑧𝐹(𝑧,𝛼)=𝐹(𝑧,𝛼)−𝐹(ğ‘§ğ‘ž,𝛼)(2.23) are called ğ¹ğ‘ž-functions.

3. Bilateral 𝒟5(ğ‘ž) as a Sum of Two Mock Theta Functions of Different Orders

(i) We shall denote the bilateral of 𝒟5(ğ‘ž) by 𝒟5,𝑐(ğ‘ž). We define it as

î€·ğ‘ž;ğ‘ž22âˆžğ’Ÿ5,𝑐(ğ‘ž)=âˆžğ‘›=âˆ’âˆžî€·ğ‘ž;ğ‘ž22ğ‘›ğ‘ž2𝑛.(3.1) Now

î€·ğ‘ž;ğ‘ž22âˆžğ’Ÿ5,𝑐(ğ‘ž)=âˆžğ‘›=âˆ’âˆžî€·ğ‘ž;ğ‘ž22ğ‘›ğ‘ž2𝑛=î“âˆžğ‘›=0î€·ğ‘ž;ğ‘ž22ğ‘›ğ‘ž2𝑛+î“âˆ’âˆžğ‘›=−1î€·ğ‘ž;ğ‘ž22ğ‘›ğ‘ž2𝑛=î“âˆžğ‘›=0î€·ğ‘ž;ğ‘ž22ğ‘›ğ‘ž2𝑛+î“âˆžğ‘›=0ğ‘ž2𝑛2+2ğ‘›î€·ğ‘ž;ğ‘ž22𝑛+1,(3.2) and we use (1.2) in the first summation and (2.2) in the second summation, to write

î€·ğ‘ž;ğ‘ž22âˆžğ’Ÿ5,𝑐(ğ‘ž)=ğ‘ž;ğ‘ž22âˆžğ’Ÿ5(ğ‘ž)+𝜔(ğ‘ž).(3.3)

Thus 𝒟5,𝑐(ğ‘ž) is a sum of a second-order mock theta function and a third-order mock theta function.

(ii) Transformation of Bilateral 𝒟5,𝑐(ğ‘ž) into bilateral 𝜔𝑐(ğ‘ž) is as follows.

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

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

2𝜓2=ğ‘Ž,𝑏𝑐,𝑑;ğ‘ž,𝑧(ğ‘Žğ‘§,𝑏𝑧,ğ‘ğ‘ž/ğ‘Žğ‘ğ‘§,ğ‘‘ğ‘ž/ğ‘Žğ‘ğ‘§;ğ‘ž)∞(ğ‘ž/ğ‘Ž,ğ‘ž/𝑏,𝑐,𝑑;ğ‘ž)∞×2𝜓2î‚¸î‚¹ğ‘Žğ‘ğ‘§/𝑐,ğ‘Žğ‘ğ‘§/ğ‘‘ğ‘Žğ‘§,𝑏𝑧;ğ‘ž,𝑐𝑑/ğ‘Žğ‘ğ‘§.(3.4) Letting ğ‘žâ†’ğ‘ž2, and setting ğ‘Ž=𝑏=ğ‘ž, 𝑐=𝑑=0, and 𝑧=ğ‘ž2 in (3.4), we get

î€·ğ‘ž;ğ‘ž22âˆžğ’Ÿ5,ğ‘î€·ğ‘ž(ğ‘ž)=3,ğ‘ž3;ğ‘ž2î€¸âˆžî€·ğ‘ž,ğ‘ž;ğ‘ž2î€¸âˆžî“âˆžğ‘›=âˆ’âˆžğ‘ž2𝑛2+2ğ‘›î€·ğ‘ž3;ğ‘ž22𝑛=î“âˆžğ‘›=âˆ’âˆžğ‘ž2𝑛2+2ğ‘›î€·ğ‘ž;ğ‘ž22𝑛+1=𝜔𝑐(ğ‘ž).(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,ğ‘ž/ğ‘Ž1,ğ‘ž/ğ‘Ž2,𝑑𝑧,ğ‘ž/𝑑𝑧;ğ‘žâˆžî€·ğ‘1,𝑐2,ğ‘ž/𝑐1,ğ‘ž/𝑐2;ğ‘žâˆž2𝜓2î‚¸ğ‘Ž1,ğ‘Ž2𝑏1,𝑏2=ğ‘ž;ğ‘ž,𝑧𝑐1𝑐1/ğ‘Ž1,𝑐1/ğ‘Ž2,ğ‘žğ‘1/𝑐1,ğ‘žğ‘2/𝑐1,𝑑𝑐1𝑧/ğ‘ž,ğ‘ž2/𝑑𝑐1𝑧;ğ‘žâˆžî€·ğ‘1,ğ‘ž/𝑐1,𝑐1/𝑐2,ğ‘žğ‘2/𝑐1;ğ‘žâˆž2𝜓2î‚¸ğ‘žğ‘Ž1/𝑐1,ğ‘žğ‘Ž2/𝑐1ğ‘žğ‘1/𝑐1,ğ‘žğ‘2/𝑐1𝑐;ğ‘ž,𝑧+idem1;𝑐2,(4.1) where 𝑑=ğ‘Ž1ğ‘Ž2/𝑐1𝑐2, |𝑏1𝑏2/ğ‘Ž1ğ‘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 2𝜓2 series on the left side of (4.1). This gives us the freedom to choose the c’s in a convenient way.

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

î€·ğ‘ž;ğ‘ž24âˆžî€·ğ‘ž4/𝑐1𝑐2;ğ‘ž2î€¸âˆžî€·ğ‘1𝑐2/ğ‘ž2;ğ‘ž2î€¸âˆžî€·ğ‘2;ğ‘ž2î€¸âˆžî€·ğ‘ž2/𝑐2;ğ‘ž2î€¸âˆžğ’Ÿ5,𝑐=ğ‘ž(ğ‘ž)2𝑐1𝑐1/ğ‘ž;ğ‘ž22âˆžî€·ğ‘ž2/𝑐2;ğ‘ž2î€¸âˆžî€·ğ‘2;ğ‘ž2î€¸âˆžî€·ğ‘1/𝑐2;ğ‘ž2î€¸âˆžî€·ğ‘ž2𝑐2/𝑐1;ğ‘ž2î€¸âˆžî“âˆžğ‘›=âˆ’âˆžî€·ğ‘ž3/𝑐1;ğ‘ž22ğ‘›ğ‘ž2𝑛𝑐+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(ğ‘ž)

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

î€·ğ‘ž;ğ‘ž22âˆžğ’Ÿ5,𝑐1(𝑧,𝛼)=(𝑧)âˆžî“âˆžğ‘›=âˆ’âˆžî€·ğ‘ž;ğ‘ž22𝑛(𝑧)ğ‘›ğ‘žğ‘›ğ›¼+𝑛.(5.1) For 𝛼=1, 𝑧=0, 𝒟5,𝑐(𝑧,𝛼) reduce to 𝒟5,𝑐(ğ‘ž).

Now

ğ·ğ‘ž,𝑧𝒟5,𝑐=1(𝑧,𝛼)𝑧𝒟5,𝑐(𝑧,𝛼)−𝒟5,𝑐=1(ğ‘§ğ‘ž,𝛼)ğ‘§î€·ğ‘ž;ğ‘ž22∞1(𝑧)âˆžî“âˆžğ‘›=âˆ’âˆžî€·ğ‘ž;ğ‘ž22𝑛(𝑧)ğ‘›ğ‘žğ‘›ğ›¼+𝑛−1(ğ‘§ğ‘ž)âˆžî“âˆžğ‘›=âˆ’âˆžî€·ğ‘ž;ğ‘ž22𝑛(ğ‘§ğ‘ž)ğ‘›ğ‘žğ‘›ğ›¼+𝑛=1ğ‘§î€·ğ‘ž;ğ‘ž22∞1(𝑧)âˆžî“âˆžğ‘›=âˆ’âˆžî€·ğ‘ž;ğ‘ž22𝑛(𝑧)ğ‘›ğ‘žğ‘›ğ›¼+𝑛(1−(1âˆ’ğ‘§ğ‘žğ‘›=1))î€·ğ‘ž;ğ‘ž22∞(𝑧)âˆžî“âˆžğ‘›=âˆ’âˆžî€·ğ‘ž;ğ‘ž22𝑛(𝑧)ğ‘›ğ‘žğ‘›ğ›¼+2𝑛=𝒟5,𝑐(𝑧,𝛼+1).(5.2) So 𝒟5,𝑐(𝑧,𝛼+1) is an ğ¹ğ‘ž-function.

Being ğ¹ğ‘ž-function it has unified properties of ğ¹ğ‘ž-functions. For example, one has the following.

(i) The inverse operator 𝐷−1ğ‘ž,𝑥 of ğ‘ž-differentiation is related to ğ‘ž-integration as

𝐷−1ğ‘ž,𝑥𝜙(𝑥)=(1âˆ’ğ‘ž)−1𝜙(𝑥)ğ‘‘ğ‘žğ‘¥.(5.3) See Jackson [16].

(ii) ğ·ğ‘›ğ‘ž,ğ‘§ğ¹ğ‘ž(𝑧,𝛼)=ğ¹ğ‘ž(𝑧,𝛼+𝑛), where 𝑛 is a nonnegative integer.

6. Behaviour of 𝒟5(ğ‘ž) outside the Unit Circle

By definition (1.1)

𝒟5(ğ‘ž)=âˆžğ‘›=0(âˆ’ğ‘ž;ğ‘ž)ğ‘›î€·ğ‘ž;ğ‘ž2𝑛+1ğ‘žğ‘›.(6.1) Replacing ğ‘ž by 1/ğ‘ž and writing 𝒟∗5(ğ‘ž) for 𝒟5(1/ğ‘ž) [10],

𝒟∗5(ğ‘ž)=âˆžğ‘›=0(−1)ğ‘›ğ‘ž(𝑛2+𝑛)/2(âˆ’ğ‘ž;ğ‘ž)ğ‘›î€·ğ‘ž;ğ‘ž2𝑛+1=1âˆ’ğ‘ž2+ğ‘ž6âˆ’ğ‘ž12+ğ‘ž20âˆ’ğ‘ž30=+â‹¯âˆžğ‘›=0(−1)ğ‘›ğ‘žğ‘›2+𝑛,(6.2) which is a 𝜃-function.

7. Lambert Series Expansion for ℎ1(ğ‘ž)

For the double series expansion, we first require the generalized Lambert series expansion for ℎ1(ğ‘ž).

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

𝒟5(ğ‘ž)=2ℎ1(ğ‘ž)−(âˆ’ğ‘ž;ğ‘ž)2âˆžğœ”(ğ‘ž),(7.1) where

ℎ1(ğ‘ž)=âˆžğ‘›=0(âˆ’ğ‘ž;ğ‘ž)2ğ‘›î€·ğ‘ž;ğ‘ž22𝑛+1ğ‘žğ‘›.(7.2) There is a slight misprint in the definition ℎ1(ğ‘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 ℎ1(ğ‘ž) is

ℎ1(ğ‘ž)=âˆžğ‘›=0(âˆ’ğ‘ž;ğ‘ž)2ğ‘›î€·ğ‘ž;ğ‘ž22𝑛+1ğ‘žğ‘›=1𝜃4(0,ğ‘ž)âˆžğ‘›=0(−1)ğ‘›ğ‘žğ‘›(𝑛+2)1âˆ’ğ‘ž2𝑛+1=12𝜃4(0,ğ‘ž)âˆžğ‘›=−∞(−1)ğ‘›ğ‘žğ‘›(𝑛+2)1âˆ’ğ‘ž2𝑛+1.(7.3)

8. ℎ1(ğ‘ž) as a Coefficient of 𝑧0 of a 𝜃-Function

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

î“âˆžğ‘Ÿ=âˆ’âˆžğ‘¥ğ‘Ÿ1âˆ’ğ‘žğ‘Ÿğ‘¦=𝐽31𝑗(𝑥𝑦,ğ‘ž)𝑗(𝑥,ğ‘ž)𝑗(𝑦,ğ‘ž)(8.1) let ğ‘žâ†’ğ‘ž2, and then put 𝑦=ğ‘ž, to get

î“âˆžğ‘Ÿ=âˆ’âˆžğ‘¥ğ‘Ÿ1âˆ’ğ‘ž2𝑟+1=𝐽32ğ‘—î€·ğ‘žğ‘¥,ğ‘ž2𝑗𝑥,ğ‘ž2î€¸ğ‘—î€·ğ‘ž,ğ‘ž2.(8.2) For |ğ‘ž|<1, and 𝑧≠0 and not an integral power of ğ‘ž, let

𝐴1(𝑧)=2𝜃4𝐽(0,ğ‘ž)32ğ‘—î€·ğ‘žğ‘§,ğ‘ž2𝑗𝑧,ğ‘ž2î€¸ğ‘—î€·ğ‘ž,ğ‘ž2î€¸ğ‘—î‚µğ‘§ğ‘ž,ğ‘ž2.(8.3)

Theorem 8.1. Let ğ‘ž be fixed with 0<|ğ‘ž|<1. Then ℎ1(ğ‘ž) is the coefficient of 𝑧0 in the Laurent series expansion of 𝐴(𝑧) in the annulus |ğ‘ž|<|𝑧|<1.

Proof. By (7.3) 2𝜃4(0,ğ‘ž)ℎ1(ğ‘ž)=âˆžğ‘›=−∞(−1)ğ‘›ğ‘žğ‘›(𝑛+2)1âˆ’ğ‘ž2𝑛+1=coefficientof𝑧0inâˆžğ‘›=âˆ’âˆžğ‘§ğ‘›1âˆ’ğ‘ž2𝑛+1î“âˆžğ‘ =−∞(−1)ğ‘ ğ‘žğ‘ 2+ğ‘ î‚µğ‘§ğ‘žî‚¶âˆ’ğ‘ =coefficientof𝑧0𝐽in32ğ‘—î€·ğ‘žğ‘§,ğ‘ž2𝑗𝑧,ğ‘ž2î€¸ğ‘—î€·ğ‘ž,ğ‘ž2î€¸ğ‘—î‚µğ‘§ğ‘ž,ğ‘ž2(8.4) dividing by 2𝜃4(0,ğ‘ž) gives the theorem.

9. An Identity for ℎ1(ğ‘ž)

Theorem 9.1. If 0<|ğ‘ž|<1 and z is neither zero nor an integral power of ğ‘ž, then 𝐴(𝑧,ğ‘ž)=𝑗𝑧,ğ‘ž2ℎ1−−1(ğ‘ž)2î‚î“âˆžğ‘Ÿ=−∞(−1)ğ‘Ÿğ‘žğ‘Ÿ2+3𝑟−1𝑧𝑟+21âˆ’ğ‘ž2𝑟+2𝑧−12î“âˆžğ‘Ÿ=−∞(−1)ğ‘Ÿğ‘žğ‘Ÿ2+3𝑟+1𝑧−𝑟−11âˆ’ğ‘ž2𝑟+2𝑧−1.(9.1) Define 1𝐿(𝑧)=−2î“âˆžğ‘Ÿ=−∞(−1)ğ‘Ÿğ‘žğ‘Ÿ2+3𝑟−1𝑧𝑟+21âˆ’ğ‘ž2𝑟+2𝑧1(9.2)𝑀(𝑧)=2î“âˆžğ‘Ÿ=−∞(−1)ğ‘Ÿğ‘žğ‘Ÿ2+3𝑟+1𝑧−𝑟−11âˆ’ğ‘ž2𝑟+2𝑧−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: ğ¹î€·ğ‘ž2𝑧=−𝑧−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 ğ¹î€·ğ‘ž2𝑧=−𝑧−1𝐹(𝑧).(9.6) We shall show that each of 𝐴(𝑧), 𝐿(𝑧), and 𝑀(𝑧) satisfies the functional equation: 𝐴1(𝑧)=2(âˆ’ğ‘ž;ğ‘ž)∞(ğ‘ž;ğ‘ž)âˆžğ½32ğ‘—î€·ğ‘žğ‘§,ğ‘ž2𝑗𝑧,ğ‘ž2î€¸ğ‘—î€·ğ‘ž,ğ‘ž2î€¸ğ‘—î‚µğ‘§ğ‘ž,ğ‘ž2,(9.7) and so ğ´î€·ğ‘ž2𝑧=12(âˆ’ğ‘ž;ğ‘ž)∞(ğ‘ž;ğ‘ž)âˆžğ½32ğ‘—î€·ğ‘ž3𝑧,ğ‘ž2î€¸ğ‘—î€·ğ‘ž2𝑧,ğ‘ž2î€¸ğ‘—î€·ğ‘ž,ğ‘ž2î€¸ğ‘—î€·ğ‘§ğ‘ž,ğ‘ž2.(9.8) We employ (2.11) on the right-hand side to get ğ´î€·ğ‘ž2𝑧=12(âˆ’ğ‘ž;ğ‘ž)∞(ğ‘ž;ğ‘ž)âˆžğ½32(−1)𝑧−1ğ‘žâˆ’1ğ‘—î€·ğ‘§ğ‘ž,ğ‘ž2(−1)ğ‘žğ‘§âˆ’1(−1)𝑧−1𝑗𝑧,ğ‘ž2î€¸ğ‘—î€·ğ‘ž,ğ‘ž2î€¸ğ‘—î‚µğ‘§ğ‘ž,ğ‘ž2î‚¶ğ´î€·ğ‘ž2𝑧=−𝑧−1𝐴(𝑧).(9.9) We now take 𝐿(𝑧): ğ¿î€·ğ‘ž2𝑧=12î“âˆžğ‘Ÿ=−∞(−1)ğ‘Ÿğ‘žğ‘Ÿ2+3𝑟−1î€·ğ‘§ğ‘ž2𝑟+21âˆ’ğ‘ž2𝑟+2î€·ğ‘§ğ‘ž2.(9.10) Writing 𝑟−1 for 𝑟on the right-hand side we have ğ¿î€·ğ‘ž2𝑧=−𝑧−1𝐿(𝑧).(9.11) Similarly only writing 𝑟+1 for 𝑟 we have ğ‘€î€·ğ‘ž2𝑧=−𝑧−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=ğ‘ž2𝑘−2 and 𝑀(𝑧) has simple poles at 𝑧=ğ‘ž2𝑘+2. Hence 𝐹(𝑧) is meromorphic for 𝑧≠0 with, at most, simple poles at 𝑧=ğ‘ž2𝑘±2.
Taking 𝑟=0 in (9.2), we calculate the residue of 𝐿(𝑧) at the point 𝑧=1/ğ‘ž2:
Residueof𝐿(𝑧)=lim𝑧→1/ğ‘ž2121ğ‘§âˆ’ğ‘ž2𝑧2ğ‘žâˆ’1𝑧−1/ğ‘ž2ğ‘žî€¸î€¸2=12ğ‘žâˆ’5.(9.13) For the residue of 𝐴(𝑧) at 𝑧=1/ğ‘ž2, take 𝑏=1, 𝑘=−1, 𝑚=2, ğ‘Ž=0 in (2.16) to get 1Residueof𝐴(𝑧)=2(âˆ’ğ‘ž;ğ‘ž)∞(ğ‘ž;ğ‘ž)âˆžğ½32𝑗1/ğ‘ž,ğ‘ž2𝑗1/ğ‘ž3,ğ‘ž2î€¸ğ‘—î€·ğ‘ž,ğ‘ž21𝐽32=12(âˆ’ğ‘ž;ğ‘ž)∞(ğ‘ž;ğ‘ž)âˆžğ‘—î€·ğ‘ž,ğ‘ž2î€¸ğ‘—î€·ğ‘ž3,ğ‘ž2î€¸ğ‘ž4ğ‘—î€·ğ‘ž,ğ‘ž2=12(âˆ’ğ‘ž;ğ‘ž)∞(ğ‘ž;ğ‘ž)∞1ğ‘ž4î€·ğ‘ž3;ğ‘ž2∞1ğ‘ž;ğ‘ž2î‚¶âˆžî€·ğ‘ž2;ğ‘ž2∞=12(âˆ’ğ‘ž;ğ‘ž)∞(ğ‘ž;ğ‘ž)∞1ğ‘ž4(1−(1/ğ‘ž))(1âˆ’ğ‘ž)ğ‘ž;ğ‘ž2î€¸âˆžî€·ğ‘ž;ğ‘ž2î€¸âˆžî€·ğ‘ž2;ğ‘ž2∞1=−2(âˆ’ğ‘ž;ğ‘ž)∞(ğ‘ž;ğ‘ž)∞1ğ‘ž5(ğ‘ž;ğ‘ž)∞(âˆ’ğ‘ž;ğ‘ž)∞1=−2ğ‘žâˆ’5.(9.14) So the residue of 𝐹(𝑧) at 𝑧=1/ğ‘ž2 is −(1/2)ğ‘ž5+0+(1/2)ğ‘ž5=0.
Now we calculate the residue at 𝑧=ğ‘ž2:
Residueof𝑀(𝑧)=limğ‘§â†’ğ‘ž212î€·ğ‘§âˆ’ğ‘ž2î€¸ğ‘žğ‘§âˆ’11âˆ’ğ‘ž2𝑧−1=limğ‘§â†’ğ‘ž212î€·ğ‘§âˆ’ğ‘ž2î€¸ğ‘žî€·ğ‘§âˆ’ğ‘ž2=ğ‘ž2,(9.15) and for the residue of 𝐴(𝑧) at 𝑧=ğ‘ž2, taking 𝑏=1, 𝑘=1, 𝑚=2, and ğ‘Ž=0 in (2.16), so 1Residueof𝐴(𝑧)=2(âˆ’ğ‘ž;ğ‘ž)∞(ğ‘ž;ğ‘ž)âˆžğ½32ğ‘—î€·ğ‘ž3,ğ‘ž2î€¸ğ‘—î€·ğ‘ž,ğ‘ž2î€¸ğ‘—î€·ğ‘ž,ğ‘ž2î€¸ğ‘ž2𝐽32=12(âˆ’ğ‘ž;ğ‘ž)∞(ğ‘ž;ğ‘ž)âˆžğ‘—î€·ğ‘ž3,ğ‘ž2î€¸ğ‘ž2=12(âˆ’ğ‘ž;ğ‘ž)∞(ğ‘ž;ğ‘ž)âˆžî€·ğ‘ž3;ğ‘ž2∞1ğ‘ž;ğ‘ž2î‚¶âˆžî€·ğ‘ž2;ğ‘ž2î€¸âˆžğ‘ž2=12(âˆ’ğ‘ž;ğ‘ž)∞(ğ‘ž;ğ‘ž)∞(1−(1/ğ‘ž))(1âˆ’ğ‘ž)ğ‘ž;ğ‘ž2î€¸âˆžî€·ğ‘ž;ğ‘ž2î€¸âˆžî€·ğ‘ž2;ğ‘ž2î€¸âˆžğ‘ž21=−2ğ‘ž.(9.16) Hence the residue of 𝐹(𝑧) at 𝑧=ğ‘ž2 is 0+(1/2)ğ‘žâˆ’(1/2)ğ‘ž=0. Hence 𝐹(𝑧) is analytic at 𝑧=ğ‘ž2.
Since 𝐹(𝑧) satisfies (9.4), so 𝐹(𝑧) is analytic at all points of the form 𝑧=ğ‘ž2𝑘±2 and hence for all 𝑧≠0.
We now apply (2.20) with 𝑛=1 and 𝑐=−1 and ğ‘ž replaced by ğ‘ž2 to get
𝐹(𝑧)=𝐹0𝑗𝑧,ğ‘ž2,(9.17) where 𝐹0 is the coefficient of 𝑧0 in the Laurent expansion of 𝐹(𝑧),𝑧≠0.
Now for |ğ‘ž|<|𝑧|<1, by Theorem 8.1, the coefficient of 𝑧0 in 𝐴(𝑧) is ℎ1(ğ‘ž).
For such 𝑧, |ğ‘ž2𝑟+2𝑧|<1 if and only if 𝑟≥0.
That is,
ğœ€î€·ğ‘ž2𝑟+2𝑧=sg(𝑟).(9.18) Hence by (2.15) 11âˆ’ğ‘ž2𝑟+2𝑧=sg(𝑟)âˆžî“ğ‘Ÿ=−∞sg(𝑟)=sg(𝑠)ğ‘ž(2𝑟+2)𝑠𝑧𝑠.(9.19) So 1𝐿(𝑧)=−2∞sg(𝑟)=sg(𝑠)sg(𝑟)(−1)ğ‘Ÿğ‘žğ‘Ÿ2+3𝑟−1+(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 ℎ1(ğ‘ž).
Hence by (9.17), we have
𝐹(𝑧)=ℎ1(ğ‘ž)𝑗𝑧,ğ‘ž2,(9.21) which gives the theorem.

10. Double Series Expansion

Now we derive the double series expansion for ℎ1(ğ‘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}, 𝑛≥0, form a Bailey pair relative to a number ğ‘Ž if

𝛽𝑛=𝑛𝑟=0𝛼𝑟(ğ‘ž)𝑛−𝑟(ğ‘Žğ‘ž)𝑛+𝑟,(10.1) for all 𝑛≥0.

Corollary 10.1 (see [5, Corollary. 2.1, page 70]). If {𝛼𝑛} and {𝛽𝑛} form a Bailey pair relative to ğ‘Ž, then î“âˆžğ‘›=0𝜌1𝑛𝜌2î€¸ğ‘›î€·ğ‘Žğ‘ž/𝜌1𝜌2î€¸ğ‘›ğ›¼ğ‘›î€·ğ‘Žğ‘ž/𝜌1î€¸ğ‘›î€·ğ‘Žğ‘ž/𝜌2𝑛=(ğ‘Žğ‘ž)âˆžî€·ğ‘Žğ‘ž/𝜌1𝜌2î€¸âˆžî€·ğ‘Žğ‘ž/𝜌1î€¸âˆžî€·ğ‘Žğ‘ž/𝜌2î€¸âˆžî“âˆžğ‘›=0𝜌1𝑛𝜌2î€¸ğ‘›î‚µğ‘Žğ‘žğ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 ğ‘Ž, 𝑏, 𝑐, and ğ‘ž be complex numbers with ğ‘Žâ‰ 1, 𝑏≠0, 𝑐≠0, ğ‘žâ‰ 0, and none ğ‘Ž/𝑏, ğ‘Ž/𝑐, ğ‘žğ‘, ğ‘žğ‘ of the form ğ‘žâˆ’ğ‘˜ with 𝑘≥0. For 𝑛≥0, define

ğ´î…žğ‘›ğ‘ž(ğ‘Ž,𝑏,𝑐,ğ‘ž)=𝑛2(𝑏𝑐)𝑛1âˆ’ğ‘Žğ‘ž2𝑛(ğ‘Ž/𝑏)𝑛(ğ‘Ž/𝑐)𝑛(1âˆ’ğ‘Ž)(ğ‘žğ‘)𝑛(ğ‘žğ‘)𝑛×𝑛𝑗=0(−1)𝑗1âˆ’ğ‘Žğ‘ž2𝑗−1(ğ‘Ž)𝑗−1(𝑏)𝑗(𝑐)ğ‘—ğ‘ž(𝑗2)(𝑏𝑐)𝑗(ğ‘ž)𝑗(ğ‘Ž/𝑏)𝑗(ğ‘Ž/𝑐)𝑗,ğµî…žğ‘›1(ğ‘Ž,𝑏,𝑐,ğ‘ž)=(ğ‘žğ‘)𝑛(ğ‘žğ‘)𝑛.(10.3) Then the sequences {ğ´î…žğ‘›(ğ‘Ž,𝑏,𝑐,ğ‘ž)} and {ğµî…žğ‘›(ğ‘Ž,𝑏,𝑐,ğ‘ž)} form a Bailey pair relative to ğ‘Ž.

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

ğ´î…žğ‘›î€·ğ‘ž2,ğ‘ž,ğ‘ž2=ğ‘ž2𝑛2+2𝑛1âˆ’ğ‘ž4𝑛+2î€¸î€·ğ‘ž;ğ‘ž22𝑛1âˆ’ğ‘ž2ğ‘žî€¸î€·3;ğ‘ž22𝑛×𝑛𝑗=0(−1)𝑗1âˆ’ğ‘ž4ğ‘—ğ‘žî€¸î€·2;ğ‘ž2𝑗−1î€·ğ‘ž;ğ‘ž22ğ‘—ğ‘žğ‘—2+ğ‘—î€·ğ‘ž2;ğ‘ž2î€¸ğ‘—î€·ğ‘ž;ğ‘ž22𝑗=(1âˆ’ğ‘ž)21+ğ‘ž2𝑛+1î€¸ğ‘ž2𝑛2+2𝑛1âˆ’ğ‘ž21âˆ’ğ‘ž2𝑛+1𝑛𝑗=0(−1)ğ‘—ğ‘žâˆ’ğ‘—2−𝑗1+ğ‘ž2𝑗=(1âˆ’ğ‘ž)21+ğ‘ž2𝑛+1î€¸ğ‘ž2𝑛2+2𝑛1âˆ’ğ‘ž21âˆ’ğ‘ž2𝑛+11+𝑛𝑗=−𝑛(−1)ğ‘—ğ‘žâˆ’ğ‘—2−𝑗,ğµî…žğ‘›î€·ğ‘ž2,ğ‘ž,ğ‘ž,ğ‘ž2=1î€·ğ‘ž3;ğ‘ž22𝑛.(10.4) Now letting ğ‘žâ†’ğ‘ž2 and then setting 𝜌1=âˆ’ğ‘ž, 𝜌2=âˆ’ğ‘ž2, ğ‘Ž=ğ‘ž2 in (10.2) we get

î“âˆžğ‘›=0ğ‘žğ‘›î€·âˆ’ğ‘ž;ğ‘ž2î€¸ğ‘›î€·âˆ’ğ‘ž3;ğ‘ž2𝑛𝛼𝑛=î€·ğ‘ž;ğ‘ž2î€¸âˆžî€·ğ‘ž4;ğ‘ž2î€¸âˆžî€·âˆ’ğ‘ž2;ğ‘ž2î€¸âˆžî€·âˆ’ğ‘ž3;ğ‘ž2î€¸âˆžî“âˆžğ‘›=0î€·âˆ’ğ‘ž;ğ‘ž2î€¸ğ‘›î€·âˆ’ğ‘ž2;ğ‘ž2î€¸ğ‘›ğ‘žğ‘›ğ›½ğ‘›.(10.5) Taking ğ´î…žğ‘› and ğµî…žğ‘› for ğ›¼î…žğ‘› and ğ›½î…žğ‘›, respectively, in (10.5) and using the definition of ℎ1(ğ‘ž), we get

î€·ğ‘ž;ğ‘ž2î€¸âˆžî€·ğ‘ž2;ğ‘ž2î€¸âˆžî€·âˆ’ğ‘ž;ğ‘ž2î€¸âˆžî€·âˆ’ğ‘ž2;ğ‘ž2∞ℎ1(ğ‘ž)=âˆžğ‘›=0ğ‘ž2𝑛2+3𝑛1âˆ’ğ‘ž2𝑛+11+𝑛𝑗=−𝑛(−1)ğ‘—ğ‘žâˆ’ğ‘—2−𝑗(10.6) or

ℎ1(ğ‘ž)=âˆ’ğ‘ž;ğ‘ž2î€¸âˆžî€·âˆ’ğ‘ž2;ğ‘ž2î€¸âˆžî€·ğ‘ž;ğ‘ž2î€¸âˆžî€·ğ‘ž2;ğ‘ž2î€¸âˆžî“âˆžğ‘›=0ğ‘ž2𝑛2+3𝑛1âˆ’ğ‘ž2𝑛+11+𝑛𝑗=−𝑛(−1)ğ‘—ğ‘žâˆ’ğ‘—2−𝑗,(10.7) which is the double series expansion for ℎ1(ğ‘ž).

This double series expansion can be used to get more properties of 𝒟5(ğ‘ž).