Math. J. Okayama Univ. 51 (2009), 121–131

SOME IDENTITIES RELATING MOCK THETA
FUNCTIONS WHICH ARE DERIVED FROM
DENOMINATOR IDENTITY
Yukari SANADA
Abstract. We show that there exists a new connection between identities satisfied by mock theta functions and special case of denominator
identities for affine Lie superalgebras.

1. Introduction
In 1920, S. Ramanujan listed 17 mock theta functions of order 3, 5 and 7
together with identities satisfied by them in his last letter to G. H. Hardy. In
the letter, he did not describe a formal definition of “order” nor prove theseidentities. However, for 3rd and 5th order, we can see that each identity
consists of mock theta functions with the same order [2], [16]. Ramanujan’s
assertion about 7th order mock theta functions is that they are not related
to each other. Some identities for order 6 , 8 and 10, which also consists ofmock theta functions with the same order, were proved by G. E. Andrews,
D. Hickerson, B. Gordon, R. J. McIntosh and Y.-S. Choi [3], [5], [6], [9]. In[13], mock theta functions 3rd order χ(q) and 6th order γ(q) are related toeach other. Recently, K. Bringmann and K. Ono use the theories of modular
forms to understand mock theta functions[4].
In this paper, we give a new view of some typical identities satisfied bymock theta functions. It is shown that these identities are obtained by
specializing the denominator identities for affine Lie superalgebras in theb 0) and B(1,b 1).
case A(1,We introduce the standard notation and some mock theta functions which
will be used in this paper.
Definition 1. Let q be a complex number such that |q| < 1. We define theq-shifted factorial for all integers n by∞Y(a; q)∞(a)∞ = (a; q)∞ :=(1 − aq i )and(a)n = (a; q)n :=.
(aq n ; q)∞i=0

For brevity, we employ the usual notation(x1 , . . . , xr )∞ = (x1 , . . . , xr ; q)∞ := (x1 )∞ · · · (xr )∞ .
Mathematics Subject Classification. Primary 00-01; Secondary 68-01.
Key words and phrases. Mock theta function, theta function, denominator identity foraffine Lie superalgebra.
121

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YUKARI SANADA

Moreover, we definej(x, q) := (x, q/x, q)∞ = (x)∞ (q/x)∞ (q)∞for all x ∈ C∗ .
By Jacobi’s triple product identity[1], [8], we havej(x, q) =∞X(−1)n qn(n−1)
2

xn .

n=−∞

Definition 2. (mock theta functions of order 3)
2∞
X
qn,f (q) :=(−q)2nn=0

χ(q) := 1 +∞X

n=1

ω(q) :=
ρ(q) :=∞
X2qnQn,m2m )
m=1 (1 − q + q

q 2n(n+1),2 )2(q;qn+1n=0∞
Xn=0

q 2n(n+1).
2m+1 + q 4m+2 )
m=0 (1 + qQn

The following mock theta functions of order 8 are found by B. Gordon
and R. J. McIntosh[9].
Definition 3. (mock theta functions of order 8)
U0 (q) :=2∞X
q n (−q; q 2 )nn=0

U1 (q) :=(−q 4 ; q 4 )n,2∞X
q (n+1) (−q; q 2 )nn=0∞
X(−q 2 ; q 4 )n+1,

22∞
q (n+1) (−q; q 2 )n X q 2n +2n+1 (−q 4 ; q 4 )n=.
V1 (q) :=2)2)(q;q(q;qn+12n+2n=0n=0In Section 1, we shall prove a following new identity for 3rd order mocktheta functions in Theorem 1.1 and then we shall show that the identity isb 1).
equal to the special case of type B(1,Theorem 1.1. For 3rd order mock theta functions ρ(q) and ω(q), the following relation holds.

MOCK THETA FUNCTIONS
22(q ; q )∞123

1(ρ(q) + ρ(−q)) + (ω(q) + ω(−q))
2=3 (q 12 ; q 12 )∞ (−q 12 ; q 24 )∞ (−q 12 ; q 24 )∞ (q 24 ; q 24 )∞.
(q 6 ; q 12 )∞In Section 2, we shall see that the identity(1.1)(q)∞ (4χ(q) − f (q)) = 3(q 3 ; q 3 )2∞(−q 3 ; q 3 )2∞b 0). This identity (1.1) is found in Ramanuis equal to the case of type A(1,jan’s last letter. We also see that some identities for mock theta functions
are derived from a specialization of the denominator identity.
We give generalized Lambert series for mock theta functions in order toprove these identities satisfied by mock theta functions. By expressing mocktheta functions with the generalized Lambert series, we can handle the mocktheta functions more easily. The following generalized Lambert series for 3rdorder mock theta functions are obtained by G. N. Watson[16].
(1.2)
(1.3)

(q)∞ f (q) = 1 + 4(q)∞ χ(q) = 1 +∞X(−1)n q n(3n+1)/2n=1∞
X

n=1(1.4)(1.5)(q 2 ; q 2 )∞ ω(q) =(q 2 ; q 2 )∞ ρ(q) =1 + qn,(−1)n (1 + q n )q n(3n+1)/2,
1 − q n + q 2n∞X(−1)n (1 + q 2n+1 )q 3n(n+1)

n=0∞
X

n=0

1 − q 2n+1,(−1)n (1 − q 4n+2 )q 3n(n+1).
1 + q 2n+1 + q 4n+2

The following generalized Lambert series for 8th order mock theta functions
are obtained by B. Gordon and R. J. McIntosh[9].
(1.6)
(1.7)
(1.8)

∞
(−q; q 2 )∞ X (−1)n q n(2n+1),U0 (q) = 2 2 2(q ; q )∞ n=−∞1 + q 4nU1 (q) =V1 (q) =∞(−q; q 2 )∞ X (−1)n q (n+1)(2n+1),(q 2 ; q 2 )∞ n=−∞1 + q 4n+22∞(−q 4 ; q 4 )∞ X (−1)n q (2n+1).
(q 4 ; q 4 )∞ n=−∞ 1 − q 4n+1

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YUKARI SANADA

We close this section by preparation of two identities as a special case ofdenominator identity for affine Lie superalgebras. We quote the denominator
identity for affine Lie superalgebras which was discovered by V. G. Kac andM. Wakimoto [10], [15]. The identity is written in the following form!
Q−α )mult(α)ρX(1−e+eQα∈∆even= e−ρε(w) w Qk.
mult(α)−α−βi )+ (1 + e)(1+eα∈∆i=1w∈Wodd+In this formula, ∆+even (resp. ∆odd ) is the set of all even (resp. odd) andpositive roots of the affine Lie superalgebra b
g and mult(α) is the dimensionof the root space b
gα and W is the Weyl group of bg and ε(w) is the signatureof w ∈ W and ρ is the Weyl vector of bg and {β1 , · · · , βk } is a maximal setof simple odd roots satisfying the inner product (βi |βj ) = 0 for all i, j =1, · · · , k (see [10], [11], [14], [15] for complete explanation and details).
b 0) isIn [15] , we can find that the denominator identity for A(1,Xeρρe R=ε(w)w,
1 + e−α2w∈Wwhere
R=∞Y(1 − e−nδ )2 (1 − e−(n−1)δ−α1 )(1 − e−nδ+α1 ),−(n−1)δ−α2 )(1 + e−nδ+α2 )(1 + e−(n−1)δ−α1 −α2 )(1 + e−nδ+α1 +α2 )(1+en=1

α1 is an even simple root, α2 is an odd simple root and δ is a primitiveimaginary root. The identity is rewritten as follows:∞Y(1 − e−nδ )2 (1 − e−(n−1)δ−α1 )(1.9) e(1 + e−(n−1)δ−α2 )(1 + e−nδ+α2 )n=1ρ(1 − e−nδ+α1 )×(1 + e−(n−1)δ−α1 −α2 )(1 + e−nδ+α1 +α2 )!
∞∞−δn(n+1) e−α1 (−n+1)−δn(n+1) enα1XXee−.
= eρ
−α2 e−nδ−α1 e−α2 e−nδ
1+e1+en=−∞n=−∞

By putting q := e−δ , x := e−α1 and y := e−α2 , (1.9) is calculated as follows:(1.10)∞∞
XX
xn+1 q n(n+1)
x−n q n(n+1)−1 + yq n
1 + xyq n
n=−∞
n=−∞

=

(q, q, x, q/x)∞.
(−y, −q/y, −xy, −q/xy)∞MOCK THETA FUNCTIONS

125

By replacing q by q 2 and substituting x = q and y = z in (1.10), we have∞X(−1)n q n(n+1)/2(q, q)∞A(z; q) :=,=n−1 q)
1+zq(−z,−z∞n=−∞(1.11)

where z 6= −q n (n ∈ Z).
b 1) isIn [15], we can also find that the denominator identity for B(1,Xeρ,eρ R =ε(w)w
1 + e−α1w∈WwhereR =∞Y(1 − e−nδ )2 (1 − e−(n−1)δ−α2 )(1 − e−nδ+α2 )(1 + e−(n−1)δ−α1 )(1 + e−nδ+α1 )(1 + e−(n−1)δ−α1 −α2 )n=1(1 − e−(n−1)δ−2α1 −2α2 )(1 − e−nδ+2α1 +2α2 )×,(1 + e−nδ+(α1 +α2 ) )(1 + e−(n−1)δ−α1 −2α2 )(1 + e−nδ+α1 +2α2 )α1 is an odd simple root and α2 is an even simple root. The identity isrewritten as follows:∞Y(1 − e−nδ )2 (1 − e−(n−1)δ−α2 )(1 − e−nδ+α2 )(1.12) eρ
(1 + e−(n−1)δ−α1 )(1 + e−nδ+α1 )(1 + e−(n−1)δ−α1 −α2 )n=1(1 − e−(n−1)δ−2α1 −2α2 )(1 − e−nδ+2α1 +2α2 )×(1 + e−(n−1)δ−α1 −2α2 )(1 + e−nδ+α1 +2α2 )(1 + e−nδ+(α1 +α2 ) )= eρ

1 2 11 2 1∞∞XXe−δ( 2 n + 2 n) e−n(α1 +α2 ) e−α2e−δ( 2 n + 2 n) e−n(α1 +α2 )−1 + e−α1 e−nδ
1 + e−α1 e−2α2 e−nδ
n=−∞
n=−∞!

.

By putting q := e−δ , x := e−α1 and y := e−α2 , (1.12) is calculated as follows:(1.13)

1 2 1
1 2 1∞∞
XX
q 2 n + 2 n (xy)nq 2 n + 2 n (xy)−n−1−n1+xq
1 + x−1 y −2 q nn=−∞n=−∞(q, q, y, q/y, x2 y 2 , q/x2 y 2 )∞.
=(−x, −q/x, −xy, −q/xy, −xy 2 , −q/xy 2 )∞By replacing q by q 2 and substituting x = zq 1/2 and y = z −1 q −1 in (1.13),
we have∞X(−1)n q n(n+1)/4B(z; q) :=(1.14)
1
1 + zq n+ 2n=−∞1=(q 2 , q)∞ (q 2 , zq, z −1 q; q 2 )∞11(−zq 2 , −z −1 q 2 )∞

,

126

YUKARI SANADA
1

where z 6= −q n+ 2 (n ∈ Z).
2. The proof of Theorem 1.1
Proof. We need the following two identities.
∞
X(−1)n q 3n(n+1)(q ; q )∞ (ω(q) + ω(−q)) = 2,
1 − q 4n+2n=−∞

(2.1)
(2.2)

222

2(q ; q )∞ (ρ(q) + ρ(−q)) = 2∞
X(−1)n q 3n(n+1).
4n+2 + q 8n+4
1+qn=−∞The identity (2.1) follows from (1.4) :∞X

1 + q 2n+1 1 − q 2n+1
(q ; q )∞ (ω(q) + ω(−q)) =(−1) q+2n+11−q
1 + q 2n+1n=0
∞4n+2 ) Xn 3n(n+1) 2(1 + q=(−1) q
1 − q 4n+222n 3n(n+1)

n=0∞
X(−1)n q 3n(n+1).
= 24n+21−qn=−∞

Similarly, the identity (2.2) follows from (1.5). Two identities (2.1) and (2.2)
are rewritten as follows:


ω(q)+
ω(−q)
(q 2 ; q 2 )∞2∞X(−1)n q 3n(n+1) 1 + q 4n+2 + q 8n+4=
×4n+21−q
1 + q 4n+2 + q 8n+4n=−∞

=∞
∞
X
X
(−1)n q 3n(n+1)(−1)n q 3n(n+1)+4n+2+1 − q 12n+61 − q 12n+6n=−∞n=−∞∞
X(−1)n q 3n(n+1)+8n+4+1 − q 12n+6n=−∞∞∞
XX(−1)n q 3n(n+1)(−1)n q 3n(n+1)+4n+2=+1 − q 12n+6
1 − q 12n+6n=−∞
n=−∞

+∞
X
(−1)−n q 3(−n)(−n+1)+8(−n)+41 − q 12(−n)+6n=−∞MOCK THETA FUNCTIONS

127

∞
∞
X
X
(−1)n q 3n(n+1)(−1)n q 3n(n+1)+4n+2=+1 − q 12n+6
1 − q 12n+6n=−∞n=−∞∞X(−1)−n q 3(−n)(−n+1)+8(−n)+4+−q −12n+6 (1 − q 12n−6 )
n=−∞

∞
∞
X
X(−1)n q 3n(n+1)+4n+2(−1)n q 3n(n+1)+=
1 − q 12n+6
1 − q 12n+6n=−∞n=−∞∞
X(−1)n q 3n(n−1)+4n−2−
1 − q 12n−6n=−∞∞∞X
X(−1)n q 3n(n+1)+4n+2(−1)n q 3n(n+1)+=
1 − q 12n+6
1 − q 12n+6n=−∞n=−∞∞
X(−1)n+1 q 3(n+1)n+4(n+1)−2−1 − q 12(n+1)−6n=−∞∞∞XX(−1)n q 3n(n+1)+4n+2(−1)n q 3n(n+1)+2=
1 − q 12n+6
1 − q 12n+6n=−∞n=−∞and(q 2 ; q 2 )∞ (ρ(q) + ρ(−q))!
∞X(−1)n q 3n(n+1)
1 − q 4n+2
= 2
×
1 + q 4n+2 + q 8n+4 1 − q 4n+2
n=−∞

∞
∞
X
X
(−1)n q 3n(n+1)(−1)n q 3n(n+1)+4n+2= 2−
1 − q 12n+6
1 − q 12n+6n=−∞
n=−∞

!

,

respectively. From these identities, the left hand side of Theorem 1.1 simplifies toω(q) + ω(−q)
2 2(2.3)(q ; q )∞ (ρ(q) + ρ(−q)) +2∞
X (−1)n q 3n(n+1)= 3
1 − q 12n+6n=−∞= 3B(−1; q 12 ).

Replacing q by q 12 and substituting z = −1 in (1.14), we have(2.4)(q 12 ; q 12 )∞ (−q 12 , −q 12 , q 24 ; q 24 )∞.
B(−1; q ) =(q 6 ; q 12 )∞12128YUKARI SANADA

By (2.3) and (2.4), the proof completes.

This proof implies that the relation in Theorem 1.1 equals to the special case of the denominator identity for affine Lie superalgebra in the caseb 1).
B(1,3. Some identities

We will prove some identities relating mock theta functions by using thedenominator identity. First we see that the identity (1.1):(q 3 ; q 3 )2∞(−q 3 ; q 3 )2∞b 0). From (1.2) and (1.3),
is derived from the denominator identity for A(1,we have∞X(−1)n q 3n(n+1)/2(3.1) (q)∞ (4χ(q) − f (q)) = 6= 6A(1; q 3 ).
3n1+qn=−∞(q)∞ (4χ(q) − f (q)) = 3

Replacing q by q 3 and substituting z = 1 in (1.11), we have
(3.2)

A(1; q 3 ) =(q 3 ; q 3 )2∞.
2 (−q 3 ; q 3 )2∞Hence, from (3.1) and (3.2), we can see that the identity (1.1) is a specialb 0).
case of the denominator identity for A(1,Next, we prove
13 (q 6 ; q 6 )2∞
2 2(q ; q )∞ ρ(q) + ω(q) =(3.3),
22 (q 3 ; q 6 )2∞
which is obtained by G. N. Watson in [16]. From (1.4) and (1.5), we have∞
3 X (−1)n q 3n(n+1)
312 2= A(−q 3 ; q 6 ).
(q ; q )∞ ρ(q) + ω(q) =6n+322 n=−∞ 1 − q2b 0),
Replacing q by q 6 and substituting z = −q 3 in (1.11) which is type A(1,we have(q 6 ; q 6 )2A(−q 3 ; q 6 ) = 3 6 ∞.
(q ; q )2∞
Hence we can prove (3.3).
Finally, we prove two identities relating mock theta functions of order 8 in
[9]:(3.4)

U0 (q) + 2U1 (q) = (−q; q 2 )3∞ (q 2 ; q 2 )∞ (q 2 ; q 4 )∞ ,MOCK THETA FUNCTIONS

(3.5)

129

V1 (q) − V1 (−q) = 2q(−q 2 ; q 2 )∞ (−q 4 ; q 4 )2∞ (q 8 ; q 8 )∞ .

From (1.6) and (1.7), we have
(3.6)

n(n+1)∞(q; q 2 )∞ X (−1)n q 2
U0 (−q) + 2U1 (−q) = 2 2 2(q ; q )∞ n=−∞ 1 + q 2n= 2(q; q 2 )∞B(q −1 ; q 2 ).
(q 2 ; q 2 )∞Replacing q by q 2 and substituting z = q −1 in (1.14), we have
(3.7)

B(q −1 ; q 2 ) =(q)2∞1.
2 (−q 2 ; q 2 )∞From (3.6) and (3.7), we prove (3.4) by replacing q by −q. Similarly, from(1.8), we have∞(−q 4 ; q 4 )∞ X (−1)n q 4n(n+1)
V1 (q) − V1 (−q) = 2q 4 4(q ; q )∞ n=−∞ 1 − q 8n+2

= 2q(−q 4 ; q 4 )∞A(−q 2 ; q 8 ).
(q 4 ; q 4 )∞Replacing q by q 8 and substituting z = −q 2 in (1.11), we haveA(−q 2 ; q 8 ) =(q 8 ; q 8 )2∞(q 8 ; q 8 )2∞=.
(q 2 ; q 8 )∞ (q 6 ; q 8 )∞(q 2 ; q 4 )∞Hence, we prove (3.5).
Remark. The denominator identities for affine Lie superalgebras in the caseb 0) and B(1,b 1) are related to Ramanujan’s summation formula 1 ψ1 series
A(1,and Bailey’s summation formula of a very-well-poised-balanced 6 ψ6 series,respectively.
4. Appendix
Here is an additional remark. We shall show that the identity in Theorem
1.1 can be seen as a relation between theta constants. From (3.3), theidentity in Theorem 1.1 becomes(4.1)

j(−q 3 , q 12 )2 + j(q 3 , q 12 )2 = 2 j(−q 6 , q 24 ) j(−q 12 , q 24 ).

Now, let v be a complex number and τ be a complex number whose
imaginary part is positive. Theta functions are defined by∞X
1 2
ϑ3 (v|τ ) = ϑ00 (v, τ ) :=en τ + nv ,2n=−∞130

YUKARI SANADA

ϑ4 (v|τ ) = ϑ01 (v, τ ) :=∞Xn=−∞e
1 21,n τ +n v+22where e(x) := e2πix [12]. Putting z = evπi and q = eτ πi , theta functions canbe rewritten as product formulas:ϑ00 (v, τ ) = j(−z 2 q, q 2 ),

ϑ01 (v, τ ) = j(z 2 q, q 2 ).

In the following relation[7, §13.23.(Transformations of the second order)](4.2)

ϑ00 (v, τ )2 + ϑ01 (v, τ )2 = 2 ϑ00 (0, 2τ ) ϑ00 (2v, 2τ ),replacing τ by 6τ and v by 32 τ and using j(x, q) = j(q/x, q) yields (4.1).
Hence, we can see that the identity in Theorem 1.1 is also the special caseof (4.2).
5. Conclusion
In [15, p195], M. Wakimoto states that “The denominator identities forthe simplest affine Lie superalgebras are Ramanujan’s mock theta functions.
In this sense, denominator identities of affine Lie superalgebras provide ageneral class of mock theta functions.” However, the specific instance for3rd or 8th order mock theta functions is not given there. We have foundthat some identities satisfied by mock theta functions are special cases ofthe denominator identity. It is plausible that these connections will assist
in giving the true meaning of mock theta functions.

Acknowledgement
The author would like to greatly thank Professor M. Ohtsuki for his kind
help, Professor M. Ito for giving some pieces of advice and Professor M.
Wakimoto for sending her his paper.
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MOCK THETA FUNCTIONS

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Yukari SANADA
Department of Mathematics,
Tsuda College
2-1-1 Tsuda-machi,
Kodaira-shi, Tokyo 187-8577,
Japan
e-mail address: sanada@tsuda.ac.jp(Received May 18, 2007 )