A Mock Theta Function of Second Order

Abstract

We consider the second-order mock theta function _...9F;5 (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 _...9F;5 (q ) outside the unit circle is a theta function and also write _h1 (q) as a coefficient of ^z0 of a theta series. First writing _h1 (q) as a coefficient of a theta function, we prove an identity for _h1 (q) .

(ProQuest: ... denotes non-US-ASCII text omitted.)

Recommended by Rodica Costin

Department of Mathematics and Astronomy, Lucknow University, Lucknow 226 007, India

Received 10 September 2009; Accepted 31 December 2009

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: [figure omitted; refer to PDF] [figure omitted; refer to PDF] and proved that ...9F;₅ (q) is a mock theta function and called it of "2nd" order.

He further showed that ...9F;₅ (q) is a sum of two mock theta functions _h1 (q) and ω(q) where _h1 (q) is of second-order and ω(q) is Ramanujan's mock theta function of third-order. This ...9F;₅ (q) will be the basis of our study in this paper.

Before we begin with the study of _...9F;5 (q) and _h1 (q) it will be appropriate to mention the work done earlier.

Gordon and McIntosh in their survey paper [8] have shown that _h1 (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,

[figure omitted; refer to PDF] where

[figure omitted; refer to PDF] Since the even part of B(q) is the ordinary theta function

[figure omitted; refer to PDF] it follows that the odd part and _h1 (q) are second-order mock theta functions. Thus ...9F;₅ (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 ...9F;₅ (q) as a bilateral q -series and show that it is also a sum of the second-order mock theta function ...9F;₅ (q) and the third-order mock theta function ω(q) . By using Bailey's transformation we have the interesting result that the bilateral ...9F;_5,c (q) is the same as the bilateral _ωc (q)_.

In Section 4, using bilateral transformation of Slater, we write ...9F;_5,c (q) as a bilateral series _ψ22 series with a free parameter c .

In Section 5, a mild generalization ...9F;_5,c (z,α) of ...9F;_5,c (q) is given and we show that this generalized function is a _Fq -function.

In Section 6 we show that ...9F;₅ (q) , outside the unit circle |q|=1 , is a theta function.

In Section 7 we state a generalized Lambert Series expansion for _h1 (q) as given in [8].

In Section 8 we show that _h1 (q) is a coefficient of ^z0 of a theta function.

In Section 9 we prove an identity for _h1 (q) using _h1 (q) as a coefficient of ^z0 of a theta function.

In Section 10 a double series expansion for _h1 (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

[figure omitted; refer to PDF] Ramanujan's mock theta function of third-order ω(q) and ν(q) is

[figure omitted; refer to PDF] [figure omitted; refer to PDF] [figure omitted; refer to PDF] We will use the following notations for θ -functions.

Definition 2.1.

If |q|<1 and x≠0, then [figure omitted; refer to PDF] If m is a positive integer and a is an integer, [figure omitted; refer to PDF] [figure omitted; refer to PDF] [figure omitted; refer to PDF] [figure omitted; refer to PDF] [figure omitted; refer to PDF] [figure omitted; refer to PDF] By Jacobi's triple product identity [12, page 282] [figure omitted; refer to PDF]

2.1. More Definitions

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

[figure omitted; refer to PDF]

If s is an integer, then

[figure omitted; refer to PDF]

Using these definitions,

[figure omitted; refer to PDF] 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 b≠0 and m≥1 . Define [figure omitted; refer to PDF] Then F is meromorphic for z≠0 , with simple poles at all points _z0 such that ^z0b =^qkm-a for some integer k . The residue of F(z) at such a point _z0 is [figure omitted; refer to PDF]

Theorem 2.3 (see [13, Theorem 1.8 (a), page 647]).

Suppose that [figure omitted; refer to PDF] for all z≠0 and that F(z) satisfies [figure omitted; refer to PDF] where 0<|q|<1 and C≠0 . Then [figure omitted; refer to PDF] Truesdell [14] calls the functions which satisfy the difference equation [figure omitted; refer to PDF] as F -function. He unified the study of these F -functions.

The functions which satisfy the q -analogue of the difference equation

[figure omitted; refer to PDF] where [figure omitted; refer to PDF] are called _Fq -functions.

3. Bilateral ...9F;₅ (q) as a Sum of Two Mock Theta Functions of Different Orders

(i) We shall denote the bilateral of _...9F;5 (q) by _...9F;5,c (q) . We define it as

[figure omitted; refer to PDF] Now

[figure omitted; refer to PDF] and we use (1.2) in the first summation and (2.2) in the second summation, to write

[figure omitted; refer to PDF]

Thus _...9F;5,c (q) is a sum of a second-order mock theta function and a third-order mock theta function.

(ii) Transformation of Bilateral _...9F;5,c (q) into bilateral _ωc (q) is as follows.

It is very interesting that the bilateral _...9F;5,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]:

[figure omitted; refer to PDF] Letting q[arrow right]^q2 , and setting a=b=q , c=d=0 , and z=^q2 in (3.4), we get

[figure omitted; refer to PDF]

4. Another Bilateral Transformation

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

[figure omitted; refer to PDF] 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 _ψ22 series on the left side of (4.1). This gives us the freedom to choose the c 's in a convenient way.

Letting q[arrow right]^q2 and setting, _a1 =_a2 =q , _b1 =_b2 =0 , and z=^q2 in (4.1), so d=^q2 /_c1c2 and 0<|z|<1 , to get

[figure omitted; refer to PDF] 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 _...9F;5,c (q)

We define the bilateral generalized function _...9F;5,c (z,α) as

[figure omitted; refer to PDF] For α=1 , z=0 , _...9F;5,c (z,α) reduce to _...9F;5,c (q).

Now

[figure omitted; refer to PDF] So _...9F;5,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 ^Dq,x-1 of q -differentiation is related to q -integration as

[figure omitted; refer to PDF] See Jackson [16].

(ii) ^Dq,zn_Fq (z,α)=_Fq (z,α+n) , where n is a nonnegative integer.

6. Behaviour of ...9F;₅ (q) outside the Unit Circle

By definition (1.1)

[figure omitted; refer to PDF] Replacing q by 1/q and writing ...9F;^5* (q) for ...9F;₅ (1/q) [10],

[figure omitted; refer to PDF] which is a θ -function.

7. Lambert Series Expansion for _h1 (q)

For the double series expansion, we first require the generalized Lambert series expansion for _h1 (q)_.

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

[figure omitted; refer to PDF] where

[figure omitted; refer to PDF] There is a slight misprint in the definition _h1 (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 _h1 (q) is

[figure omitted; refer to PDF]

8. _h1 (q) as a Coefficient of ^z0 of a θ -Function

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

[figure omitted; refer to PDF] let q[arrow right]^q2 , and then put y=q, to get

[figure omitted; refer to PDF] For |q|<1 , and z≠0 and not an integral power of q , let

[figure omitted; refer to PDF]

Theorem 8.1.

Let q be fixed with 0<|q|<1 . Then _h1 (q) is the coefficient of ^z0 in the Laurent series expansion of A(z) in the annulus |q|<|z|<1 .

Proof.

By (7.3) [figure omitted; refer to PDF] dividing by 2_θ4 (0,q) gives the theorem.

9. An Identity for _h1 (q)

Theorem 9.1.

If 0<|q|<1 and z is neither zero nor an integral power of q , then [figure omitted; refer to PDF] Define [figure omitted; refer to PDF] [figure omitted; refer to PDF] [figure omitted; refer to PDF] The scheme will be first to show that F(z) satisfies the functional relation: [figure omitted; refer to PDF] 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 [figure omitted; refer to PDF] We shall show that each of A(z) , L(z) , and M(z) satisfies the functional equation: [figure omitted; refer to PDF] and so [figure omitted; refer to PDF] We employ (2.11) on the right-hand side to get [figure omitted; refer to PDF] We now take L(z) : [figure omitted; refer to PDF] Writing r-1 for r on the right-hand side we have [figure omitted; refer to PDF] Similarly only writing r+1 for r we have [figure omitted; refer to PDF] 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=^q2k-2 and M(z) has simple poles at z=^q2k+2 . Hence F(z) is meromorphic for z≠0 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 :

[figure omitted; refer to PDF] For the residue of A(z) at z=1/^q2 , take b=1 , k=-1 , m=2 , a=0 in (2.16) to get [figure omitted; refer to PDF] 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 :

[figure omitted; refer to PDF] and for the residue of A(z) at z=^q2 , taking b=1 , k=1 , m=2 , and a=0 in (2.16), so [figure omitted; refer to PDF] 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 z≠0.

We now apply (2.20) with n=1 and c=-1 and q replaced by ^q2 to get

[figure omitted; refer to PDF] where _F0 is the coefficient of ^z0 in the Laurent expansion of F(z), z≠0 .

Now for |q|<|z|<1 , by Theorem 8.1, the coefficient of ^z0 in A(z) is _h1 (q).

For such z , |^q2r+2 z|<1 if and only if r≥0 .

That is,

[figure omitted; refer to PDF] Hence by (2.15) [figure omitted; refer to PDF] So [figure omitted; refer to PDF] 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 (q).

Hence by (9.17), we have

[figure omitted; refer to PDF] which gives the theorem.

10. Double Series Expansion

Now we derive the double series expansion for _h1 (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 } , n≥0 , form a Bailey pair relative to a number a if

[figure omitted; refer to PDF] for all n≥0 .

Corollary 10.1 (see [5, Corollary. 2.1, page 70]).

If {_αn } and {_βn } form a Bailey pair relative to a , then [figure omitted; refer to PDF] 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 a≠1 , b≠0 , c≠0 , q≠0 , and none a/b , a/c , qb , qc of the form ^q-k with k≥0 . For n≥0 , define

[figure omitted; refer to PDF] Then the sequences {^{An[variant prime]} (a,b,c,q)} and {^{Bn[variant prime]} (a,b,c,q)} form a Bailey pair relative to a .

Letting q[arrow right]^q2 and then taking a=^q2 , b=c=q , in (10.3), we get

[figure omitted; refer to PDF] Now letting q[arrow right]^q2 and then setting _ρ1 =-q , _ρ2 =-^q2 , a=^q2 in (10.2) we get

[figure omitted; refer to PDF] Taking ^{An[variant prime]} and ^{Bn[variant prime]} for ^{αn[variant prime]} and ^{βn[variant prime]} , respectively, in (10.5) and using the definition of _h1 (q) , we get

[figure omitted; refer to PDF] or

[figure omitted; refer to PDF] which is the double series expansion for _h1 (q) .

This double series expansion can be used to get more properties of ...9F;₅ (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, R. J. McIntosh, "Modular transformations of Ramanujan's fifth and seventh order mock theta functions," The Ramanujan Journal , vol. 7, pp. 193-222, 2003.

[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.

[5] G. E. Andrews, D. Hickerson, "Ramanujan's "lost" notebook. VII. The sixth order mock theta functions," Advances in Mathematics , vol. 89, no. 1, pp. 60-105, 1991.

[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.

[7] B. Gordon, R. J. McIntosh, "Some eighth order mock theta functions," Journal of the London Mathematical Society , vol. 62, no. 2, pp. 321-335, 2000.

[8] B. Gordon, R. J. McIntosh, "A survey of classical mock theta functions," preprint

[9] R. J. McIntosh, "Second order mock theta functions," Canadian Mathematical Bulletin , vol. 50, no. 2, pp. 284-290, 2007.

[10] K. Hikami, "Transformation formula of the "second" order mock theta function," Letters in Mathematical Physics , vol. 75, no. 1, pp. 93-98, 2006.

[11] G. E. Andrews, "Mordell integrals and Ramanujan's lost notebook," Analytic Number Theory, Lecture Notes , vol. 899, pp. 10-48, Springer, Berlin, 1981.

[12] G. H. Hardy, E. M. Wright An Introduction to the Theory of Numbers , Oxford University Press, London, UK, 1968., 4th.

[13] D. Hickerson, "A proof of the mock theta conjectures," Inventiones Mathematicae , vol. 94, no. 3, pp. 639-660, 1988.

[14] C. Truesdell An Essay Toward a Unified Theory of Special Functions , of Annals of Mathematics Studies, no. 18, pp. iv+182, Princeton University Press, Princeton, NJ, USA, 1948.

[15] G. Gasper, M. Rahman Basic Hypergeometric Series , vol. 35, of Encyclopedia of Mathematics and Its Applications, pp. xx+287, Cambridge University Press, Cambridge, UK, 1990.

[16] F. H. Jackson, "Basic integration," Quart. J. Math. ( Oxford), (2) , vol. 2, pp. 1-16, 1951.

[17] G. E. Andrews, B. C. Berndt Ramanujan's Lost Notebook, Part I , pp. xiv+437, Springer, New York, NY, USA, 2005.

[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.