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TRANSACTIONS OF THEAMERICAN MATHEMATICAL SOCIETYVolume 363, Number 1, January 2011, Pages 439–455S 0002-9947(2010)05181-5Article electronically published on August 31, 2010

KAC-WAKIMOTO CHARACTERS

AND UNIVERSAL MOCK THETA FUNCTIONS

AMANDA FOLSOM

Abstract. In recent work, Bringmann and Ono answer a question of Kac andshow that character formulas for s�(r + 1, 1)∧ modules due to Kac and Waki-moto are “holomorphic parts” of nonholomorphic modular functions. Here,we confirm a speculation of Ono that these characters are, up to a simpleq-series, the universal mock theta functions g2(ω, q) and g3(ω, q) of Gordonand McIntosh. Using recent work of Bringmann-Ono, Kang, Zwegers, andGordon-McIntosh, we show that g2(ω; q) and g3(ω; q) are, up to classical thetafunctions and η-products, the characters of Kac and Wakimoto. As a con-sequence, we include a “dictionary” that gives a character formula for everyclassical mock theta function of Ramanujan, as well as subsequent naturalgeneralizations.

1. Introduction and statement of results

“Monstrous moonshine” is one of the most revered results connecting modularforms and Lie algebras. Conjectured by Conway and Norton [4] in 1979 and provedby Borcherds [1] in 1992, monstrous moonshine relates the Fourier coefficients ofthe modular j-function, the normalized hauptmodul for SL2(Z), to dimensions ofirreducible representations of the Monster. In his proof, Borcherds constructs ageneralized Kac-Moody Lie algebra M , and reveals the connection to the j-functionvia a denominator identity satisfied by M .

Preceding Moonshine, however, is the fundamental work of V. Kac [6], who firstexhibits a link between infinite-dimensional Lie algebras and modular forms. In [6],Kac proves the so-called Weyl-Kac character formula and denominator identity,and interprets the latter in the special case of affine Lie algebras as “Macdonaldidentities”. The key idea that combinatorial identities can be obtained by compar-ing two different calculations of characters of representations of infinite-dimensionalLie algebras was introduced by Kac in this paper.

Another example relating modular forms and Lie algebras is the fundamentalwork of Lepowsky and Milne [12, 13], 1978. Here, Lepowsky and Milne relate theRogers-Ramanujan functions, essentially modular forms of weight 0, to Weyl-KacLie algebras. More specifically, as a consequence of [12, Theorem 5.1], Lepowskyand Milne show that the Rogers-Ramanujan identities, one of which is the following

Received by the editors April 21, 2009 and, in revised form, May 4, 2009.2000 Mathematics Subject Classification. Primary 11F22, 11F37, 17B67, 11F50.

c©2010 American Mathematical Society

439

440 AMANDA FOLSOM

identity between a basic hypergeometric-type series and an infinite product:∑

n≥0

qn2

(q; q)−1n =

∏

n≥1

(1− q5n−1)−1(1− q5n−4)−1,(1.1)

where (a; q)n :=∏n−1

j=0 (1 − aqj), become principally specialized characters for the

standard modules for A(1)1 , an infinite-dimensional generalized Cartan matrix Lie

algebra. In subsequent work in 1981, Lepowsky and Wilson [14] formulate an“abstract” Rogers-Ramanujan identity, which in special cases coincides with theclassical identities. There are many other examples of this phenomenon, for examplethe Virasoro algebras, as well as certain classical infinite-dimensional affine Liealgebras. (See for example the book of Kac [7].)

The more recent work of Kac and Wakimoto [9, 10, 18] beginning in the mid-1990s has also made significant connections between number theory and affine Liesuperalgebras. Of note along our line of discussion thus far is the 2000 work ofD. Zagier [19], who reformulates and proves, in terms of combinatorial modularforms, a conjectured denominator identity of Kac and Wakimoto. Another objectof study in the work of Kac and Wakimoto is the specialized character formulafor trL(Λ(s);r+1)q

L0 , where L(Λ(s); r + 1) is the irreducible s�(r + 1, 1)∧ module

with highest weight Λ(s), and L0 the “energy operator”, or Hamiltonian. In [3],Bringmann and Ono answer a question of Kac regarding the “modularity” of thecharacter formulas trL(Λ(s);r+1)q

L0 , and show that these character formulas are the“holomorphic parts” of nonholomorphic modular functions.

In addition to exhibiting modular properties, many q-series arising from suchLie algebras also have combinatorial properties. For example, the left hand side ofthe first Rogers-Ramanujan identity (1.1) may be interpreted as the combinatorialgenerating function for the number of partitions of integers with minimal difference2, and the right hand side of (1.1) the generating function for partitions whose partsare congruent to 1 or 4 mod 5. This type of result also holds more generally withrespect to Virasoro algebras. Namely, one finds in the work of Kac and Wakimoto[8] and Rocha-Caridi [17] that the generating function for partitions whose partssatisfy congruence relations (an infinite product analogous to the right hand sideof (1.1)) is equal to a character formula:

chi,k(q) = q(hi,k−ck24 ) ·

∏

1≤n�≡0,±i (mod 2k+1)

(1− qn)−1.

In light of these results, Ono speculated that the character formulas of Kac andWakimoto are related to the universal mock theta functions of Gordon and McIn-tosh. We show that this is indeed the case.

The universal mock theta functions g2(ω; q) and g3(ω; q) of Gordon and McIn-tosh [5] are generalizations of the original mock theta functions of Ramanujan.The original mock theta functions are basic hypergeometric-type series appearingin Ramanujan’s notebooks and his last letter to Hardy. The functions g2(ω; q) andg3(ω; q) are aptly named “universal”, as it is shown by Gordon and McIntosh [5]that all classical mock theta functions of Ramanujan and subsequent natural gen-eralizations may be written in terms of either g2(ω; q) or g3(ω; q), depending onthe parity of their “order”, a number assigned to each mock theta function. Bothclassically and in their more modern generalizations, mock theta functions havebeen objects of extensive research in the areas of number theory, combinatorics,

CHARACTERS AND MOCK THETA FUNCTIONS 441

and mathematical physics. We refer the reader to [16] and [5] for a more detailedaccount. The universal mock theta functions are defined by

g2(ω; q) :=∑

n≥0

(−q; q)nq12 (n

2+n)

(ω; q)n+1(qω−1; q)n+1,

g3(ω; q) :=∑

n≥0

qn2+n

(ω; q)n+1(qω−1; q)n+1.

Kang, in [11], considers the universal mock theta functions g2(ω; q) and g3(ω; q),and shows that both may be expressed in terms of the “mock Jacobi forms”μ(u, v; τ ) studied by Zwegers in his important Ph.D. thesis [20] under the direc-tion of D. Zagier. For a precise definition of μ(u, v; τ ) and more on the work andresults of Kang and Zwegers, see §2 below. In addition to results of Bringmannand Ono [3] and Gordon and McIntosh [5], we will use the work of Kang [11] andZwegers [20] in the course of the proof of our main theorem, Theorem 1.1 below.

Theorem 1.1. Let q := e2πiτ , τ ∈ H. The following are true.

(i) For r ∈ N and s ∈ Z we have

g2(q

r4−

s2 ; q

r2

)= Θ−1

r (τ )trL(Λ(s);r+1)qL0 − ηr,s(τ ).

(ii) For r ∈ 3N and s ∈ Z we have

g3(q

r9−

s3 ; q

r3

)

= Θ−1r (τ )

(q

r36+

s6 ·trL(Λ(s);r+1)q

L0+q−r36−

s6 ·trL(Λ

(s+ r3)

;r+1)qL0

)−βr,s(τ ).

(iii) For r ∈ 6N and s ≡ r6 (mod r) we have

g3(q

r9−

s3 ; q

r3

)= Θ−1

r (τ ) · q r36+

s6 · trL(Λ(s);r+1)q

L0 − ψr,s(τ ).

The functions Θr(τ ), ηr,s(τ ), βr,s(τ ), and ψr,s(τ ) in ( i), ( ii), and ( iii) are sumsof modular forms defined in (2.14), (2.15), (2.20) and (2.24).

By subtracting the identity in Theorem 1.1 (iii) from that in (ii), we see that withhypotheses on r and s the Kac-Wakimoto character formula is a sum of modularforms.

Corollary 1.2. Let r ∈ 6N and s ≡ r6 (mod r). Then we have

Θ−1r (τ ) · q− r

36−s6 · trL(Λ

(s+ r3 )

;r+1)qL0 = βr,s(τ )− ψr,s(τ ).

Example 1. The third order mock theta function of Ramanujan,

f(q) :=∑

n≥0

qn2

(−q; q)2n,

has been of particular interest. Notable recent works include those of Bringmannand Ono [2] and Zwegers [20]. Using formula (3.11) of [5] which relates f(q) to theuniversal mock theta function g3(ω; q), we find, after making appropriate substi-tutions in Theorem 1.1 (ii), that the third order mock theta function f(q) can be

442 AMANDA FOLSOM

expressed in terms of the Kac-Wakimoto character formulas as follows:

f(−q) = −4qΘ−112 (τ )

(q

12 · trL(Λ(1),13)q

L0 + q−12 · trL(Λ(5);13)q

L0

)+ F (τ ).(1.2)

Here, F (τ ) is a sum of modular forms defined by

F (τ) := 4qβ12,1(τ) +(q2; q2)7∞

(q; q)3∞(q4; q4)3∞.

Here and throughout, (a; q)∞ := limn→∞(a; q)n.

Example 2. As a second example, we consider the order 10 mock theta functionappearing in Ramanujan’s lost notebook:

φ(q) :=∑

n≥0

q12n(n+1)

(q; q2)n+1.

Using the expression for φ(q) in terms of the universal mock theta function g2(ω; q)found in [5, (5.18)] and applying Theorem 1.1 (i), we find

φ(q) = 2q · Θ−110 (τ ) · trL(Λ(1);11)q

L0 +Φ(τ ),

where

Φ(τ ) := iq58 · (q

10; q10)2∞(q5; q5)∞

·ϑ(2τ + 1

2 ; 5τ)

ϑ(2τ ; 10τ )− 2q · η10,1(τ ).

Remark 1. In the Appendix (§3), we provide a “dictionary” that gives a characterformula for every classical mock theta function of Ramanujan, as well as subsequentnatural generalizations. The identities are derived from Theorem 1.1 in a mannersimilar to that described in Example 1 and Example 2.

2. Proof of Theorem 1.1

In this section we will prove Theorem 1.1. In the course of our proof we willmake use of results due to Bringmann and Ono [3] in their study of the Kac-Wakimoto character formulas trL(Λ(s);r+1)q

L0 , results due to Gordon-McIntosh [5]

on the universal mock theta functions, work of Kang [11] who relates the universalmock theta functions to Zwegers’ mock Jacobi forms μ(u, v; τ ), as well as the workof Zwegers [20] on the mock Jacobi forms μ(u, v; τ ).

We begin with the work of Kac and Wakimoto [10], which implies for r ∈ N, s ∈Z,

trL(Λ(s);r+1)qL0 = 2q

r−124 − s

2 · η2(2τ )

ηr+3(τ )· Lr,s(τ ),(2.1)

where

Lr,s(τ ) :=∑

k=(k1,k2,...,kr)∈Zr

q12

∑i ki(ki+1)

1 + q−s+∑

i ki.

In [3], Bringmann and Ono explicitly relate the character formulas in (2.1) to weight0 nonholomorphic forms. A key identity used in their work, which we will also makeuse of here, is the following [3, Theorem 2.1].


Proposition (Bringmann-Ono). For integers r ≥ 1 and s ∈ Z, we have

Lr,s(τ ) =∑

d=(d1,d2,...,dr)∈Dr

ϑd(τ )μdr,r,s(τ ).(2.2)

The sum in (2.2) is a finite lattice sum over Dr := {d = (d1, d2, . . . , dr) ∈ Zr | 0 ≤

di ≤ i− 1 for each 1 ≤ i ≤ r}. The functions ϑd(τ ) and μdr,r,s(τ ) are defined by

ϑd(τ ) :=

r−1∏

i=1

ϑ(di,−di+1, i, i+ 1; τ ),(2.3)

μdr,r,s(τ ) := q−dr2 +

d2r2r

∑

n∈Z

qrn2 (n+1)−dn

1 + qrn−d−s,(2.4)

where

ϑ(a, b, c, d; τ ) := qcd2 (

ac +

bd−

12 )

2

ϑ

((ad+ bc− cd

2

)τ − 1

2; cdτ

),(2.5)

and the function ϑ(z; τ ) defining (2.5) is the classical Jacobi theta function

ϑ(z; τ ) :=∑

n∈Z

eπi(n+12 )

2τ+2πi(n+ 1

2 )(z+12 ).(2.6)

(The empty product in (2.3) is understood to equal 1.) One also understands themodularity of the function μdr,r,s(τ ). An observation of Bringmann and Ono in [3]reveals that

q−s2μdr,r,s(τ ) = −iq

d2r2r ϑ

(1

2− drτ ; rτ

)μ

(1

2− (s+ dr)τ,

1

2− drτ ; rτ

),(2.7)

where the function μ(u, v; τ ) is nearly a weight 1/2 Jacobi form studied by Zwegersin [20], who shows that a correction term may be added to the function μ(u, v; τ )to produce weight 1/2 harmonic weak Maass forms at torsion points. The functionμ(u, v; τ ) is defined by

μ(u, v; τ ) :=e(u/2)

ϑ(v; τ )

∑

n∈Z

(−1)ne(nv)qn(n+1)/2

1− e(u)qn,

where e(α) := e2πiα. Of the fundamental properties and transformations associatedto μ(u, v; τ ) (given in [20]), we will make use of the following.

Proposition (Zwegers). Let u, v, u+ z, v + z ∈ C \ Zτ + Z, τ ∈ H. Then

μ(u+ z, v + z; τ ) = μ(u, v; τ ) +iη3(τ )ϑ(u+ v + z; τ )ϑ(z; τ )

ϑ(u; τ )ϑ(v; τ )ϑ(u+ z; τ )ϑ(v + z; τ ).(2.8)

The function η(τ ) in (2.8) is the Dedekind η–function, a classical weight 1/2modular form defined by

η(τ ) := q124

∞∏

n=1

(1− qn).

444 AMANDA FOLSOM

Recalling (2.2) and (2.7), we begin by applying (2.8) with

u :=(r2− s

)τ,

v :=rτ

2,

z :=1

2−(dr +

r

2

)τ

to the μ-function appearing in (2.7), and find

μ

(1

2− (s+ dr)τ,

1

2− drτ ; rτ

)=μ

((r2− s

)τ,

rτ

2; rτ

)+ ψr,s,dr

(τ ),(2.9)

where

ψr,s,dr(τ ) :=

iη3(rτ )ϑ(12 +

(r2 − s− dr

)τ ; rτ

)ϑ(12 −

(dr +

r2

)τ ; rτ

)

ϑ((

r2−s

)τ ; rτ

)ϑ(rτ2 ; rτ

)ϑ(12−(s+ dr)τ ; rτ

)ϑ(12−drτ ; rτ

) .(2.10)

Next, we let

S1(r, s; τ ) := −iqs2

∑

(d1,...,dr)∈Dr

qd2r2r ϑ

(1

2− drτ ; rτ

)ϑd(τ )μ

((r2− s

)τ,

rτ

2; rτ

),

(2.11)

S2(r, s; τ ) := −iqs2

∑

(d1,...,dr)∈Dr

qd2r2r ϑ

(1

2− drτ ; rτ

)ϑd(τ )ψr,s,dr

(τ ).

(2.12)

By (2.2) and (2.9) we have

Lr,s(τ ) = S1(r, s; τ ) + S2(r, s; τ ).(2.13)

Next, we define

Θr(τ ) := 2qr6−

124Θr(τ ),(2.14)

ηr,s(τ ) := iq−r4+

s2 · η4(rτ )

η2(rτ2

)ϑ((

r2 − s

)τ ; rτ

)(2.15)

⎧⎪⎪⎨

⎪⎪⎩

− q−r8

2 · η4(rτ)η2(2rτ) ·

ϑ( 12−

rτ2 ;rτ)ϑ( 1

2+(r2−s)τ ;rτ)

ϑ( 12−sτ ;rτ)ϑ(( r

2−s)τ ;rτ)ϑ( rτ2 ;rτ)

, sr ∈ 1

2 + Z,

+q−s2−

r8 ·Θ−1

r (τ ) η2(2τ)ηr+3(τ) · S2(r, s; τ ),

sr �∈ 1

2 + Z,

where the function Θr(τ ) is the weight −1/2 modular form defined in [3] by

Θr(τ ) :=η2(2τ )

ηr+3(τ )·

∑

d=(d1,d2,...,dr)∈Dr

ϑd(τ ) · qd2r2r ϑ

(1

2− drτ ; rτ

),(2.16)

and S2(r, s; τ ) is the finite sum of modular forms defined in (2.12). We note thatin [3] it is pointed out that

Θr(τ ) = −2rη2r+2(2τ )

η2r+3(τ ).

Proof of Theorem 1.1 (i). The proof of Theorem 1.1 (i) will now follow from Propo-sition 2.1 below, after a straightforward calculation using (2.1), (2.2), (2.12), (2.13),(2.14) and (2.15).


Proposition 2.1. For r ∈ N and s ∈ Z, we have that

S1(r, s; τ ) = qr8+

s2 · η

r+3(τ )

η2(2τ )·Θr(τ ) · g2

(q

r4−

s2 ; q

r2

)

+ iqs−r8 ·Θr(τ ) ·

η4(rτ )ηr+3(τ )

η2(2τ )η2(rτ2

)ϑ((

r2 − s

)τ ; rτ

) ,(2.17)

and for r ∈ N, s ∈ Z such that sr ∈ 1

2 + Z, we have that

S2(r, s; τ ) = −qs2

2· η

4(rτ )ηr+3(τ )

η2(2τ )η2(2rτ )·

ϑ(12 − rτ

2 ; rτ)ϑ(12 +

(r2 − s

)τ ; rτ

)

ϑ(12 − sτ ; rτ

)ϑ((

r2 − s

)τ ; rτ

)ϑ(rτ2 ; rτ

)

·Θr(τ ).

(2.18)

�

Proof of Theorem 1.1 (ii). To prove Theorem 1.1 (ii), we use the following relationbetween the universal mock theta functions g2(ω; q) and g3(ω; q) due to Gordonand McIntosh [5].

Proposition (Gordon-McIntosh). The following identity is true:

g3(ζ4; q4) = qζ−2g2(ζ

6q; q6) + q−1ζ2g2(ζ6q−1; q6)− J(ζ; q),(2.19)

where ζ = e2πiz, and

J(ζ; q) := ζ4q74 · (q

2; q2)3∞(q12; q12)∞(q4; q4)∞(q6; q6)2∞

· ϑ(2z + τ ; 2τ )ϑ(12z + 6τ ; 12τ )

ϑ(4z; 2τ )ϑ(6z − τ ; 2τ ).

Next we define

βr,s(τ ) := J(qr36−

s12 ; q

r12 ) + q

r36+

s6 · ηr,s(τ ) + q−

r36−

s6 ηr,s+ r

3(τ ).(2.20)

In (2.19), we replace q by qr12 and then let ζ = q

r36−

s12 . With these substitutions,

(2.19) becomes

g3(q

r9−

s3 ; q

r3

)= q

r36+

s6 · g2

(q

r4−

s2 ; q

r2

)+ q−

r36−

s6 · g2

(q

r4−

s+ r3

2 ; qr2

)

− J(qr36−

s12 ; q

r12 ).

(2.21)

Theorem 1.1 (ii) will now follow after a short calculation from Theorem 1.1 (i) using(2.21) and (2.20). �

Proof of Theorem 1.1 (iii). From the work of Gordon and McIntosh [5], one canalso deduce another relation between the universal mock theta functions g2(ω1; q)and g3(ω2; q) under suitable hypotheses on the arguments ω1 and ω2. Namely, from[5, Equation (7.1)] and the expressions given for the functions u2 and u3 following(6.9) on page 37 of [5], one may deduce the following:

g3(x(q)4; q4) = −iq−

12T (x(q), q)

ϑ(4τ ; 12τ )+ qx(q)−2g2(x(q)

6q; q6),(2.22)

where x(q) = q16+n, n ∈ Z, and T (x(q), q) is a theta function defined in [15].

To prove (iii), we first replace q by qr12 in (2.22). With this change of variable,

we see that x(q)6q is replaced by qr6+

rn2 , and q6 is replaced by q

r2 . Thus, with these

substitutions, we find the universal mock theta function g2(q

r6+

rn2 ; q

r2

)in (2.22).

446 AMANDA FOLSOM

If we set r6 + rn

2 = r4 − s

2 , we see that because we require s, n ∈ Z and r ∈ N, wemust have r ∈ 6N and s ≡ r

6 (mod r). Combining these observations, we deducefor r ∈ 6N and s ≡ r

6 (mod r) that

g3(q

r9−

s3 ; q

r3

)= q

r36+

s6 g2

(q

r4−

s2 ; q

r2

)− iq−

r24T(q

r36−

s12 , q

r12

)

ϑ(rτ3 ; rτ

) .(2.23)

Next we define

ψr,s(τ ) := iq−r24T(q

r36−

s12 , q

r12

)

ϑ(rτ3 ; rτ

) + qr36+

s6 ηr,s(τ ).(2.24)

Theorem 1.1 (iii) now follows after combining (2.23) with Theorem 1.1 (i).

Proof of Proposition 2.1 (2.17). We observe that (2.17) will follow after applyingto the function S1(r, s; τ ) defined in (2.11) the following proposition due to Kang[11], which relates the universal mock theta function g2(ω; q) to Zwegers’ functionsμ(u, v; τ ).

Proposition (Kang). Let α ∈ C be such that α �∈ Z+ 2τZ. Then

ie(α)g2(e(α); q) =η4(2τ )

η2(τ )ϑ(2α; 2τ )+ e(α)q−

14μ(2α, τ ; 2τ ).(2.25)

More specifially, one lets α = 12

(r2 − s

)τ in (2.25), and replaces τ by rτ/2. The

identity given in (2.17) follows after a short calculation. �

Proof of Proposition 2.1 (2.18). The proof of (2.18) requires more work. We firstobserve that many of the functions defining S2(r, s; τ ) are independent of the indicesof summation d = (d1, d2, . . . , dr) ∈ Dr. To this end we define

S2(r, s; τ ) :=∑

d∈Dr

qd2r2r ϑd(τ )

ϑ(12 +

(r2 − s− dr

)τ ; rτ

)ϑ(12 −

(dr +

r2

)τ ; rτ

)

ϑ(12 − (s+ dr)τ ; rτ

) ,

(2.26)

so that

S2(r, s; τ ) =q

s2 η3(rτ )

ϑ((

r2 − s

)τ ; rτ

)ϑ(rτ2 ; rτ

) · S2(r, s; τ ).(2.27)

Here, our goal is to rewrite the sum S2(r, s; τ ), not as a finite lattice sum overelements (d1, d2, . . . , dr) ∈ Dr, but as a visible product of modular forms. Webegin by using the well-known product expansion for the Jacobi theta functionsϑ(ω; τ ), known as the Jacobi triple product identity:

ϑ(ω; τ ) = −iq18 e−πiω

∏

n≥1

(1− qn)(1− e(ω)qn−1)(1− e(−ω)qn).(2.28)

Next, we let

Fr,s(q, z)

:=∏

n≥1

(1 + qr(n−1)+ r2−sz)(1 + qrn−

r2+sz−1)(1 + qr(n−1)− r

2 z)(1 + qrn+r2 z−1)

(1 + qr(n−1)−sz)(1 + qrn+sz−1),

(2.29)


so that after applying (2.28) to (2.26), we have

S2(r, s; τ ) = −qr8

∏

n≥1

(1− qrn)∑

d=(d1,d2,...,dr)∈Dr

qd2r2r + dr

2 ϑd(τ )Fr,s(q, q−dr).(2.30)

We next establish the following series expansion for the function Fr,s(q, z). �

Lemma 2.2. For r ∈ N, s ∈ Z such that sr ∈ 1

2 +Z, the function Fr,s(q, z) satisfies

Fr,s(q, z) = −qr24

2·ϑ(12 +

(r2 − s

)τ ; rτ

)ϑ(12 − rτ

2 ; rτ)

ϑ(12 − sτ ; rτ

) · 1

η2(2rτ )

∑

n∈Z

qrn(n−1)

2 zn.

(2.31)

Proof. We first observe that

Fr,s(q, zqr)

(2.32)

=∏

n≥1

(1 + qrn+r2−sz)(1 + qr(n−1)− r

2+sz−1)(1 + qrn−r2 z)(1 + qr(n−1)+ r

2 z−1)

(1 + qrn−sz)(1 + qr(n−1)+sz−1)

=∏

n≥2

(1 + qr(n−1)+ r2−sz)(1 + qr(n−1)− r

2 z)

(1 + qr(n−1)−sz)

∏

n≥0

(1 + qrn−r2+sz−1)(1 + qrn+

r2 z−1)

(1 + qrn+sz−1)

=(1 + q−sz)(1 + q−

r2+sz−1)(1 + q

r2 z−1)

(1 + qr2−sz)(1 + q−

r2 z)(1 + qsz−1)

Fr,s(q, z)

= z−1Fr,s(q, z).

Next, we wish to use the relation (2.32) to write a power series expansion forFr,s(q, z) in the variable z. We note that Fr,s(q, z) as a function in z has polesfor z = −qrn+s, n ∈ Z. By making the assumption that s

r ∈ 12 + Z, we see that

for each pole there is a corresponding zero of the same order, and thus Fr,s(q, z) isholomorphic. For example, if s

r ∈ 12+Z

≥0, each of the poles z = −qrn+s, n ≥ 1, will

be canceled by the zero given by z = −qrmn+r2 , wheremn := n+ s

r−12 ≥ 1

2+sr , which

is an integer ≥ 1. One can argue similarly that each of the poles z = −qrn+s, n ≤ 0,will be canceled by other zeros of the function Fr,s(q, z). The case s

r ∈ − 12 + Z

≤0

follows by a similar argument. Thus, with these assumptions on r and s, if we writethe power series expansion for Fr,s(q, z) in z as

Fr,s(q, z) =∑

n∈Z

an(q)zn,(2.33)

then the relation (2.32) implies

an(q) · qrn = an+1(q).(2.34)

448 AMANDA FOLSOM

After iterating (2.34), one finds that an(q) = qrn(n−1)

2 a0(q) for all n ∈ Z. Thus, wehave

Fr,s(q, z) = a0(q)∑

n∈Z

qrn(n−1)

2 zn.(2.35)

To determine a0(q), we see using (2.29) that

Fr,s(q, 1) = −q−r8+

r24ϑ(12 +

(r2 − s

)τ ; rτ

)ϑ(12 − rτ

2 ; rτ)

ϑ(12 − sτ ; rτ

)η(rτ )

.(2.36)

On the other hand, using (2.35) we see that

Fr,s(q, 1) = q−r8 a0(q)

∑

n∈Z

qr2 (n+

12 )

2

(2.37)

= 2a0(q)∏

n≥1

(1− qrn)(1 + qrn)2,(2.38)

where the product expansion in (2.38) is determined by writing the sum defining(2.37) in terms of the Jacobi theta function ϑ(z; τ ), applying (2.28), and simplifying.Combining (2.38) with (2.36) implies

a0(q) = −qr24

2·ϑ(12 +

(r2 − s

)τ ; rτ

)ϑ(12 − rτ

2 ; rτ)

ϑ(12 − sτ ; rτ

) · 1

η2(2rτ ).(2.39)

Substituting(2.39) into (2.35) finishes the proof. �

To continue, we apply the series expansion for Fr,s(q, z) obtained in (2.31) to

the expression for S2(r, s; τ ) given in (2.30), replacing z by q−d. Once again, manyof the functions and terms appearing are independent of the indices of summationd = (d1, d2, . . . , dr) ∈ Dr. One finds that the only sum left to be understood (with

respect to S2(r, s; τ )) is

∑

d=(d1,d2,...,dr)∈Dr

qd2r2r + dr

2 ϑd(τ )∑

n∈Z

qrn(n−1)

2 −drn.(2.40)

A short calculation reveals

−q−r8−

dr2 ϑ

(1

2− drτ ; rτ

)=

∑

n∈Z

qrn(n−1)

2 −drn.(2.41)

Applying (2.41) to (2.40) shows that the sum in (2.40) may be written as

−q−r8

∑

d=(d1,d2,...,dr)∈Dr

qd2r2r ϑd(τ )ϑ

(1

2− drτ ; rτ

).(2.42)

It is explained in [3] that the sum in (2.42) is equal to

−q−r8ηr+3(τ )

η2(2τ )Θr(τ ).

In summary, one begins with the expression in (2.27) for S2(r, s; τ ) and sub-stitutes (2.35) into (2.30), using the expression for a0(q) derived above in (2.39).

Finally, one substitutes for (2.42) the expression −q−r8ηr+3(τ)η2(2τ) Θr(τ ). After simpli-

fying, one finds (2.18). �


3. Appendix: Character formulas for the mock theta functions

In this section, we give a list of character formulas for all classical mock thetafunctions as well as subsequent natural generalizations. To derive them, we applyTheorem 1.1 to the “mock theta conjectures”. The mock theta conjectures are nolonger conjectures, but are now known identities due to G. Andrews, F. Garvan,D. Hickerson, B. Gordon, R. McIntosh and Y.-S. Choi. The conjectures may beexpressed as formulas relating each mock theta function to the universal mocktheta functions g2(ω; q) and g3(ω; q). We follow the notation and presentation ofthe mock theta conjectures as they appear in [5, (3.4), (3.5), (3.10), (3.11), (5.2),(5.10), (5.12), (5.18)], and refer the reader there for a complete account. Theauxiliary functions G(q), H(q), θ4(0, q), ψ(q), and j(x, q) appearing are defined inTable 8 at the end of this section. We remark that there exist mock theta functionsother than those listed in Tables 1 - 7. However it is unnecessary to list theircharacter formulas here: they are easily derived using the formulas provided inTables 1 - 7, as well as “linear relations” between these mock theta functions andthose listed below. (See [5].)

450 AMANDA FOLSOM

Table 1. Character formulas for order 3 mock θ-functions

f(−q) = −4q · Θ−112 (q

12 · trL(Λ(1);13)q

L0 + q−12 · trL(Λ(5);13)q

L0)

+ 4qβ12,1(τ ) +(q2;q2)7∞

(q;q)3∞(q4;q4)3∞

φ(q) = −2q · Θ−112 (q

12 · trL(Λ(1);13)q

L0 + q−12 · trL(Λ(5);13)q

L0)

+ +2qβ12,1(τ ) +(q2;q2)7∞

(q;q)3∞(q4;q4)3∞

ψ(q) = q · Θ−112 (q

12 · trL(Λ(1);13)q

L0 + q−12 · trL(Λ(5);13)q

L0)− qβ12,1(τ )

χ(−q) = −q ·Θ−112 (q

12 ·trL(Λ(1);13)q

L0+q−12 ·trL(Λ(5);13)q

L0)

+ qβ12,1(τ )+(q4;q4)3∞(q6;q6)3∞

(q2;q2)2∞(q3;q3)∞(q12;q12)2∞

ω(q) = Θ−16 (trL(Λ(−1);7)q

L0 + trL(Λ(1);7)qL0)− β6,−1(τ )

v(q) = −q · Θ−112 (trL(Λ(−2);13)q

L0 + trL(Λ(2);13)qL0) + qβ12,−2(τ ) +

(q4;q4)3∞(q2;q2)2∞

ρ(q) = − 12 · Θ−1

6 (trL(Λ(−1);7)qL0 + trL(Λ(1);7)q

L0) + 12 β6,−1(τ )

+ 32

(q6;q6)4∞(q2;q2)∞(q3;q3)2∞

ξ(q) = 1 + 2q2 · Θ−118 · trL(Λ(3);19)q

L0 − 2qψ18,3(τ )

σ(−q) = q2 ·Θ−136 (q

32 ·trL(Λ(3);37)q

L0+q−32 ·trL(Λ(15);37)q

L0)

− q2β36,3(τ )+(q2;q2)3∞(q12;q12)3∞

(q;q)∞(q4;q4)2∞(q6;q6)2∞



f0(q) = −2q2 · Θ−130 (q

32 · trL(Λ(4);31)

qL0 + q−32 · trL(Λ(14);31)

qL0 )

+ 2q2β30,4 + θ4(0, q5)G(q)

F0(q) = 1 + q · Θ−115 (q

34 · trL(Λ(2);16)

qL0 + q−34 · trL(Λ(7);16)

qL0 )

− qβ15,2 − qψ(q5)H(q2)

φ0(−q) = −q · Θ−115 (q

34 · trL(Λ(2);16)

qL0 + q−34 · trL(Λ(7);16)

qL0 )

+ qβ15,2 + j(−q2, q5)G(q2)

ψ0(q) = q2 · Θ−130 (q

32 · trL(Λ(4);31)

qL0 + q−32 · trL(Λ(14);31)

qL0 )

− q2β30,4 + qj(q, q10)H(q)

χ0(q) = 2 + 3q · Θ−115 (q

34 · trL(Λ(2);16)

qL0 + q−34 · trL(Λ(7);16)

qL0 )

− β15,2 − j(q2, q5)G2(q)

f1(q) = −2q3 · Θ−130 (q

12 · trL(Λ(−2);31)

qL0 + q−12 · trL(Λ(8);31)

qL0 )

+ 2q3β30,−2 + θ4(0, q5)H(q)

F1(q) = q · Θ−115 (q

14 · trL(Λ(−1);16)

qL0 + q−14 · trL(Λ(4);16)

qL0 )

− qβ15,−1 + ψ(q5)G(q2)

φ1(−q) = q2 · Θ−115 (q

14 · trL(Λ(−1);16)

qL0 + q−14 · trL(Λ(4);16)

qL0 )

− −q2β15,−1 − qj(−q, q5)H(q2)

ψ1(q) = q3 · Θ−130 (q

12 · trL(Λ(−2);31)

qL0 + q−12 · trL(Λ(8);31)

qL0 )

− q3β30,−2 + j(q3, q10)G(q)

χ1(q) = 3q · Θ−115 (q

14 · trL(Λ(−1);16)

qL0 + q−14 · trL(Λ(4);16)

qL0 )

− 3qβ15,−1 + j(q, q5)H2(q)

452 AMANDA FOLSOM


F0(q) = 2 + 2q · Θ−121 (q

54 · trL(Λ(4);22)q

L0 + q−54 · trL(Λ(11);22)q

L0)

− 2qβ21,4(τ )− j2(q3, q7)(q; q)−1∞

F1(q) = 2q2 · Θ−121 (q

34 · trL(Λ(1);22)q

L0 + q−34 · trL(Λ(8);22)q

L0)

− 2q2β21,1(τ ) + qj2(q, q7)(q; q)−1∞

F2(q) = 2q2 · Θ−121 (q

14 · trL(Λ(−2);22)q

L0 + q−14 · trL(Λ(5);22)q

L0)

− 2q2β21,−2(τ ) + j2(q2, q7)(q; q)−1∞


A(q2) = q · Θ−18 trL(Λ(2);9)

qL0 − qη8,2(τ)− q(−q2; q2)∞(−q4; q4)2∞(q8; q8)∞

B(q) = Θ−14 trL(Λ(0);5)

qL0 − η4,0(τ)

μ(q4) = −2q · Θ−14 trL(Λ(0);5)

qL0 + 2qη4,0(τ) +(q2; q2)∞(q4; q4)3∞(q8; q8)∞

(q; q)2∞(q16; q16)212


φ(q4) = −2q · Θ−112 trL(Λ(4);13)q

L0

+ 2qη12,4(τ ) +(q2;q2)3∞(q3;q3)2∞(q12;q12)3∞

(q;q)2∞(q6;q6)3(q8;q8)∞(q24;q24)∞

ψ(q4) = −q3 · Θ−112 trL(Λ(0);13)q

L0 + q3η12,0(τ )

+ q3(q2;q2)2∞(q4;q4)∞(q24;q24)2∞

(q;q)∞(q3;q3)(q8;q8)2∞



S0(−q2) = −2q · Θ−116 trL(Λ(6);17)q

L0 + 2qη16,6(τ ) +j(−q, q2)j(q6, q16)

j(q2, q8)

S1(−q2) = −2q · Θ−116 trL(Λ(2);17)q

L0 + 2qη16,2(τ ) +j(−q, q2)j(q2, q16)

j(q2, q8)


φ(q) = 2q · Θ−110 trL(Λ(1);11)q

L0 − 2qη10,1(τ ) +(q10; q10)2∞j(−q2, q5)

(q5; q5)∞j(q2, q10)

ψ(q) = 2q · Θ−110 trL(Λ(3);11)q

L0 − 2qη10,3(τ )− q(q10; q10)2∞j(−q, q5)

(q5; q5)∞j(q4, q10)

X(−q2) = −2q · Θ−140 trL(Λ(18);41)q

L0 + 2q · Θ−140 trL(Λ(2);41)q

L0

+2qη40,18(τ )− 2qη40,2(τ )

+(q4;q4)2∞(j(−q2,q20)j(q12,q40)+2q(q40;q40)3)

(q2;q2)∞(q20;q20)∞(q40;q40)∞j(q8,q40)

χ(−q2) = −2q3 · Θ−140 trL(Λ(14);41)q

L0 − 2q5 · Θ−140 trL(Λ(6);41)q

L0

+2q3η40,14(τ ) + 2q5η40,6(τ )

+q2(q4;q4)2∞(2q(q40;q40)3∞−j(−q6,q20)2j(q4,q40))

(q2;q2)∞(q20;q20)∞(q40;q40)∞j(q16,q40)

454 AMANDA FOLSOM

Table 8. Auxiliary functions

G(q) :=

∞∏

n=1

(1− q5n−4)−1(1− q5n−1)−1

H(q) :=

∞∏

n=1

(1− q5n−2)−1(1− q5n−3)−1

θ4(0, q) := (q; q)∞(q; q2)∞

ψ(q) :=(q2; q2)∞(q; q2)∞

j(x, q) := (x; q)∞(x−1q; q)∞(q; q)∞

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Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706

E-mail address: [email protected] address: Department of Mathematics, Yale University, New Haven, Connecticut 06520E-mail address: [email protected]








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