UNIMODAL SEQUENCES AND STRANGE FUNCTIONS: A FAMILY OF QUANTUM MODULAR FORMS

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1 UNIMODAL SEQUENCES AND STRANGE FUNCTIONS: A FAMILY OF QUANTUM MODULAR FORMS KATHRIN BRINGMANN, AMANDA FOLSOM, AND ROBERT C RHOADES Abstract In this paper, we construct an infinite family of quantum modular forms from combinatorial rank moment generating functions for strongly unimodal sequences The first member of this family is Kontsevich s strange function studied by Zagier These results rely upon the theory of mock Jacobi forms As a corollary, we exploit the quantum and mock modular properties of these combinatorial functions in order to obtain asymptotic expansions 1 Introduction and Statement of Results A sequence of integers {a j } s j=1 is called a strongly unimodal sequence of size n if there exists an integer k such that 11 0 < a 1 < a < < a k > a k+1 > > a s > 0 and a a s = n A number of familiar sequences are strongly unimodal, for example, the sequence of binomial coefficients { n j 1 } n+1 j=1 with n even Attached to strongly unimodal sequences is a notion of rank, analogous to the well known notion of the rank of an integer partition For more on partition ranks, see for example original works by Atkin and Swinnerton-Dyer [9], Dyson [19], and Ramanujan [7], and the more recent joint work of the first author and Ono [15] related to mock modular forms The rank of a strongly unimodal sequence is equal to s k+1, the number of terms after the maximal term minus the number of terms that precede it For example, there are six strongly unimodal sequences of size 5: {5}, {1, 4}, {4, 1}, {1,, 1}, {, }, {, } Their respective ranks are 0, 1, 1, 0, 1, 1 By letting w resp w 1 keep track of the terms after resp before a maximal term, we have that um, n, the number of size n and rank m sequences, satisfies 1 Uw; q := n=1 m= um, n w m q n = wq; q n w 1 q; q n q n+1, where for n N 0, w; q n := n 1 j=0 1 wqj Recently, Bryson, Ono, Pitman, and the third author [16] studied this function in the special case w = 1, namely 1 U1; q = n=1 m= 1 m um, nq n = n=0 u e n u o n q n, n=1 1 Note The function Uw; q, given in 1, is equal to the function U w; q as defined in [16] 1

2 KATHRIN BRINGMANN, AMANDA FOLSOM, AND ROBERT C RHOADES where u e n resp u o n denotes the number of unimodal sequences of size n with even resp odd rank They showed that for every root of unity ζ, U1; ζ = F ζ 1, where Kontsevich s strange function is defined by F q := q; q n Prior, Zagier [1] proved that this function satisfies the identity 1 F q = 1 1 n q n 1 4, n n=1 n=0 where is the Kronecker symbol Neither side of the identity 1 makes sense simultaneously Indeed, the right hand side of 1 converges in the unit disk q < 1, but nowhere on the unit circle The identity 1 means that at roots of unity ζ, F ζ which is clearly a finite sum agrees with the limit as q approaches ζ radially within the unit disk of the function on the right hand side of 1 Moreover, Zagier proved that for x Q \ {0} 14 φx + ix φ 1 i i = w + x ηwdw, x π 0 n=1 1 eπinw is the Dedekind eta-function where φx := e πix 1 F e πix and ηw := e πiw 1 Note that the constant i/π in 14 is given explicitly in [16] There, the authors also gave a new proof of 14, using the fact that U1; q is a weak mixed mock modular form for q < 1 Here, we slightly modify the definition of mixed mock modular form given in [18] to mean functions that lie in the tensor product of the general spaces of mock modular forms and weakly holomorphic modular forms up to possible rational multiples of q powers In particular, we do not require as in [18] these functions to be holomorphic at the cusps Weak mixed mock modular forms in this sense occur in a variety of areas including combinatorics [], algebraic geometry [9], Lie theory [], Joyce invariants [5], and quantum black holes [18, 4] The similarity between 14 and the usual modular transformation formula of a modular form in part motivated Zagier [] to introduce the notion of a quantum modular form A weight k 1 Z quantum modular form is a complex-valued function f on Q, such that for all γ = a c d b SL Z, the function h γ : Q \ γ 1 C defined by ax + b 15 h γ x := fx εγcx + d k f cx + d satisfies a suitable property of continuity or analyticity The εγ in 15 are suitable complex numbers, such as those in the theory of half-integral weight modular forms when k 1Z \ Z This paper gives an infinite family of quantum modular forms from the moments of the unimodal rank statistic In general, such moment functions are of both number theoretic and combinatorial interest For example, in their celebrated work [8], Atkin and Garvan discovered a partial differential equation relating the bivariate generating functions for the partition statistics rank and crank, leading to exact linear relations between rank and crank

3 A FAMILY OF QUANTUM MODULAR FORMS moments In [4], Andrews provided a beautiful combinatorial interpretation of partition rank moments in terms of k-marked Durfee symbols Andrews also discovered a relationship between partition rank moments and the smallest parts partition statistic in [5], which has led to further work by Garvan [], for example In addition to intrinsic combinatorial interest, moment functions have been shown to satisfy modular properties For example, works including [, 1, 1] exhibit relationships to weak Maass forms and mock theta functions To state our results, we define for r N 0 the following weighted moment functions, where throughout q := e πi 16 φ r := πi r+1 1 m um, nq r m, n 1 n=1 m Z and Q r X, Y Q[X, Y ] is the polynomial 17 Q r X, Y := 0 µ r 0 l r µ with rational coefficients c r µ, l defined in 19 normalized, with Y Y 1 are given by 4 Q 0 X, Y 1 =, 4 Q 1 X, Y 1 = 4X + Y, 4 Q X, Y 1 4 Q X, Y 1 = c r µ, lx l Y µ Q[X, Y ], q n 1 4, For example, the first few polynomials = 4 10X + 5X + 6Y + 180XY + 108Y, 105 7X + 140X + 154X + Y + 40XY + 160X Y + 10Y + 50XY + 70Y Note that in particular the first member of the family φ r is up to a constant the strange function studied by Zagier and Kontsevich discussed above That is, φ 0 = πiq 1 4 U1; q = πiφ It is not difficult to see that the functions φ r may also be written in terms of the twisted unimodal moment functions u r, defined for integers r 0 by u r q := 1 m um, nm r q n n=1 m Z The moments m um, nmr of the unimodal rank statistic are analogous with the rank and crank partition moments, functions which have drawn wide combinatorial interest since Atkin and Garvan famously introduced them [8] There is a vast literature on such objects, including asymptotic questions and congruence properties While the unimodal rank moments are exponentially large for even r [14], it is surprising that the twisted moments m 1m um, nm r, as a consequence of our results, are only polynomially large in n We have chosen to handle the more complicated expressions m 1m um, nq r m, n 1/4 because the generating functions for these numbers have a fixed weight as modular objects

4 4 KATHRIN BRINGMANN, AMANDA FOLSOM, AND ROBERT C RHOADES as seen in Theorem 11, while the generating function for the twisted moments will have a mixed weight To relate these generating functions φ r to the twisted unimodal moments u r, by symmetry, we note that u r+1 q = 0 for integers r 0 In particular, using 16, we find that 18 where we define 19 φ r = πi r+1 c r µ, l := 0 µ r 0 l r µ c r µ, l πi µ µ u µ l qq 1 4, l+1 6 µ Γ 1 + r µ Γ 1 + r µ!l!r µ l + 1! Q The coefficients c r µ, l are indeed in Q, as it is well known for integers k N 0, that Γ 1 + k π Q The twisted moment functions also naturally extend the unimodal function U1; q discussed above; namely, u 0 q = U1; q = q 1 4 πi 1 φ 0 To state our first result, we define another polynomial 110 P r X, Y := b r N, M X N+1 Y M, 0 N r 0 M r where the coefficients b r N, M are given explicitly in 14 Our first theorem establishes that the unimodal moment functions φ r are quantum modular forms on Q\{0}, and that their transformation law also extends to H The function H r below is defined in 15 Theorem 11 Let r N 0 If H Q\{0}, we have that φ r i r φ r 1 = P r w, i e πiw R sinh πw coshπw dw + H r, where H r = 0 for Q\{0} In particular, the functions φ r are quantum modular forms Remarks 1 The transformation law given in 111 in the case H essentially establishes the mock modular properties of the unimodal rank moment functions φ r In the course of proving 111 in the case Q \ {0}, we show that for each integer r 0, the function φ r is defined for Q Moreover, in Theorem 51 of 5, we pay special attention to the case r = 1, and establish an explicit finite value for φ 1 h k h, k Z as the value of a polynomial in the root of unity e πih/k Our functions naturally arise from mock Jacobi forms It would be interesting to investigate whether a theory of quantum Jacobi forms could be developed that contains functions arising in this paper as special cases Our next theorem exploits the automorphic properties given in Theorem 11, and establishes the asymptotic behavior of the moment functions u r While such properties are of independent interest, we also point out that these functions are related to the quantum moment functions φ r by 18 To describe their asymptotic behavior, we use the Bernoulli

5 A FAMILY OF QUANTUM MODULAR FORMS 5 polynomials B k x and Euler polynomials E k x, defined by the generating functions respectively ze xz e z 1 = k=0 e xz e z + 1 = k=0 B k x zk k!, E k x zk k!, Theorem 1 For non-negative integers r, as t 0 +, we have that e πt 1 ur e πt = r+1 πt k r n B n r + 1 k! n In particular, we have that k=0 e πt 1 ur e πt 6r r n r 5 B r+1 + B r+1 6 E r+1+k n 5 6 The paper is organized as follows In Section we provide relevant background information on modular forms, Jacobi forms, and mock Jacobi forms, as well as Bernoulli and Euler polynomials In Section we prove Theorem 11, and in Section 4 we establish Theorem 1 In Section 5 we pay special consideration to the moment function φ 1 Acknowledgements The research of the first author was supported by the Alfried Krupp Prize for Young University Teachers of the Krupp foundation and the research leading to these results has received funding from the European Research Council under the European Union s Seventh Framework Programme FP/ / ERC Grant agreement n 50 - AQSER The second author is grateful for the support of National Science Foundation CAREER grant DMS-15815, and the Max Planck Institute for Mathematics in Bonn, Germany The authors also thank the referee for his or her helpful comments and suggestions Preliminaries Here, we provide preliminary information on automorphic forms in 1, and Bernoulli and Euler polynomials in 1 Automorphic forms In this section, we recall some fundamental properties of certain modular and mock Jacobi forms We start with the well known transformation law for the Dedekind η-function Lemma 1 For γ = a b c d SL Z, we have that 1 η γ = χ γ c + d 1 η, where χ γ is a 4th root of unity, which can be given explicitly in terms of Dedekind sums [6] In particular, we have that η 1 = iη,

6 6 KATHRIN BRINGMANN, AMANDA FOLSOM, AND ROBERT C RHOADES Here and throughout the square root is defined by the principal branch of the logarithm Moreover, we require the usual Jacobi theta function, defined for z C and H, by ϑz; := e πiν+πiνz+ 1 ν 1 +Z This function is well known to satisfy the following transformation law [6, 801 and 808] Lemma For λ, µ Z and γ = a b c d SL Z, we have that In particular, we have that ϑz + λ + µ; = 1 λ+µ q λ e πiλz ϑz;, z ϑ c + d ; γ = χ γ c + d 1 πicz e c+d ϑz; z ϑ ; 1 = i ie πiz ϑ z; The Jacobi theta function also satisfies the well known triple product identity w = e πiz ϑz; = iq 1 8 w 1 1 q n 1 wq n 1 1 w 1 q n n=1 Additionally, we require the following classical Taylor expansion see for example [0]: ϑz; = πz η exp G k πizk k! Here for even integers k, the Eisenstein series are defined by G k := B k k + σ k 1 nq n, where σ l n := d n dl and B k denotes the kth Bernoulli number We also make use of Zwegers s functions A l z 1, z ; [4] see also [7] and [10], defined for l N, H, z C, and z 1 C \ Z + Z, by 4 A l z 1, z ; := e lπiz 1 n Z n=1 k=1 1 ln q lnn+1 e πinz 1 q n e πiz 1 These functions may be completed into non-holomorphic Jacobi forms by setting k=0 Â l z 1, z ; := A l z 1, z ; + R l z 1, z ; The non-holomorphic completions of these higher level Appell functions are defined by R l z 1, z ; := i l 1 ekz 1 ϑ z + k + l 1 ; l R lz 1 z k l 1 ; l, where ex := e πix and where = u + iv Rz; := sgnn E n 1 +Z n + Imz v v 1 n 1 q n e πinz,

7 A FAMILY OF QUANTUM MODULAR FORMS 7 with Ez := z 0 e πt dt Proposition below shows that the so-called error to modularity of the function Rz; is the Mordell integral, defined for z C and H by 5 hz; := R e πiw πzw coshπw dw Proposition Zwegers, [] For z C and H, we have that Rz + 1; = Rz;, z R ; 1 = ie πiz Rz; + hz; Moreover, the completed higher level Appell functions A l z 1, z ; transform as follows Proposition 4 Zwegers, [4] For n 1, n, m 1, m Z and γ = a b c d SL Z, we have  l z 1 + n 1 + m 1, z + n + m ; = 1 ln 1+m 1 ez 1 ln 1 n n 1 z q ln 1 z1 c lz  l = c + de 1 + z 1 z c + d, z c + d ; γ c + d  l z 1, z ; n 1n  l z 1, z ;, We further require dissection properties of the functions ϑ and R see [11, 8, 4] Lemma 5 With notation as above, we have for n N ϑ z; n 1 = q l n 1 n 1 πil n e z+ 1 ϑ nz + l n 1 + n 1 ; n, n l=0 R z; n 1 = n l=0 q l n n 1 n 1 πil e z+ 1 R nz + l n 1 + n 1 ; n Bernoulli and Euler polynomials In this section, we recall certain properties of the Bernoulli polynomials B k x and Euler polynomials E k x, defined in 11 and 11, respectively, as well as their special values 1 B k := B k 0, E k := k E k One property we make use of is a dissection property of the Bernoulli polynomials Namely, for m N 0 + 1, it is well known that see Chapter of [1] 6 m 1 B k mx = m k 1 j=0 a=0 B k x + a m Another splitting property that we use is the following: x + y k k 7 k B k = B j xe k j y, j which follows easily from the definition of the Euler and Bernoulli polynomials, using the fact that z e x+yz = zexz e z 1 e z 1 e yz e z + 1

8 8 KATHRIN BRINGMANN, AMANDA FOLSOM, AND ROBERT C RHOADES Here and throughout, we let ζ N := e πi/n for N N The next lemma expresses derivatives of secant in terms of Euler polynomials Lemma 6 With notation as above, we have that for c N 0 π sec c+1 = 1 c 5 6 c+1 E c+1 6 Proof This follows quickly from Theorem of [17] Namely, using the facts that E c = E c 1 5 6, and Ec 1 1 = 0, gives the claim A fourth property that we use expresses the Euler numbers as integrals known see 18 on p4 of [1] for example that for k N 0, w k 8 R coshπw dw = i k E k Note that E k 1 = 0 for k N Proof of Theorem 11 Namely, it is Here, we ultimately conclude Theorem 11 from Proposition 6, Proposition 7, and Proposition 8 below In 1, we establish properties of mock Jacobi forms related to the unimodal rank generating function, and in, we construct mock modular forms from its Taylor coefficients In, we establish quantum modularity and prove Theorem 11 Until otherwise indicated, throughout this section, we take H 1 Mock Jacobi forms and unimodal ranks Here we establish properties of mock Jacobi forms associated to the unimodal rank generating function We begin by writing Uw; q in terms of the Appell functions A l u, v; defined in 4 Throughout, for w 1, w C, we let Uw 1 ; w := Uew 1 ; ew Lemma 1 Let w = ez With notation as above, we have that 1 Uz; = A1 z, z; w 1 A z, ; w 1 w 1 q; q Proof Entry 47 of Ramanujan s lost notebook see p67 of [6] gives with a = w, b = w 1 that Uz; equals 1 q n 1 1 n q nn+1 w n w 1 w 1 wq; q n w 1 q; q n 1 w 1 q; q 1 wq n n=0 We note that second sum on the right hand side of 1 is easily seen to be equal to 1 A 1 z, z; w 1 w 1 q; q Using these facts, the result follows after applying the identity [9] 1 q n 1 1 = A 1 w 1 1 w wq; q n w 1 z, ; q; q n w 1 w 1 q; q n=0 n Z

9 A FAMILY OF QUANTUM MODULAR FORMS 9 Next we define a normalization of the function Uz; Y + z; := w 1 w 1 q 1 4 Uz; = η 1 w 1 A z, ; A 1 z, z;, where the second equality follows from Lemma 1 Using Proposition, we now establish a transformation law for Y +, which is a key step in showing quantum modularity of the functions φ r To state this, we define H z; := i where hz; is given in 5, and gz; := i ϑz; h z; gz;, η R e πiw πwz sinh πw cosh πw dw Proposition With notation as above, we have that ie πiz z Y + ; 1 1 i Y + z; = H z; To prove Proposition we rather work with a second normalization of the function Uz;, namely X + z; := e πz v w 1 w 1 q; q Uz; = w 1 A z, ; A 1 z, z; e πz v Moreover we need the completed function Xz; := w 1 Â z, ; Â1z, z; e πz v = Â z, 0; Â1z, z; e πz v, where the second equality in follows from the first transformation in Proposition 4 Using Proposition 4, it is not difficult to establish the following modularity result for Xz; Proposition With notation as above, for γ = a c d b SL Z, we have that z X c + d ; γ = c + d Xz; From Proposition, we can establish the following transformation property of X + z; Proposition 4 With notation as above, we have that z X + ; 1 1 X + i i z; = ϑ z; h z; + η ± ±h z ± 1 ; e πz v Proof Using Proposition we obtain that z X + ; 1 1 X + z; i = f 1 z; + f z;,

10 10 KATHRIN BRINGMANN, AMANDA FOLSOM, AND ROBERT C RHOADES with f 1 z; :=ϑ 1 ; e πz v 1 ± ϑ ; e πz v ±e ± πiz z R ± 1 ; ±e ±πiz R z ;, z z f z; :=ϑ ; 1 R ; 1 e πz v 1 ϑ z; R z; e πz v We next simplify f 1 and f Firstly, using Lemma and Proposition, we obtain that 4 f z; = ϑ z; h z; e πz v Next Lemma and Proposition yield that ϑ 1 ; e πz v 1 ±e ± πiz z R ± 1 ; ± = 1 πz e v ϑ 1 ; ± R z ± 1 ; + h z ± 1 ; Now Lemma 5, the fact that ϑ0; = 0, and Proposition, give that ϑ 1 ; π = i sin q 1 6 ϑ ;, R z ± 1 ; = q 1 6 e πiz± 1 R z ; + R z; q 1 6 e πiz± 1 R z + ; ± ± Thus and hence 5 R z ± 1 ; π = i sin q 1 6 ±e ±πiz R z ;, ± f 1 z; = i q 1 6 ϑ ; ±h z ± 1 ; e πz v ± ± Combining 4, 5, and the fact that ϑ; = iq 1 6 η gives the claim Proof of Proposition First note that ±h z ± 1 ; = i gz; ± The result now follows immediately from Proposition 4 and Lemma 1, using the fact that Y + z; = e πz v η X+ z;

11 A FAMILY OF QUANTUM MODULAR FORMS 11 Taylor coefficients and unimodal ranks Next, using the results from 1, we construct mock modular forms from the Taylor coefficients of the unimodal rank generating function The fuctions Hz; and Y + z; are holomorphic in z, and it is not difficult to see that they are both odd functions in z, so we may write 6 Y + z; = a r z r+1, 7 H z; = h r z r+1 The next lemma describes the modularity properties of the Taylor coefficients a r of Y + z; Lemma 5 With notation as above, we have that a r 1 i r = π r j r j! 1j+1 i j r a j + h j 0 j r Proof Proposition directly yields z Y + ; 1 = ie πiz i Y + z; + H z; Inserting the Taylor expansions 6 and 7, and Taylor expanding the exponential function, gives that l a r 1 z r+1 πiz = i i a j + h j z j+1 l! = i i l=0 z r+1 0 j r Equating the coefficients of z r+1 gives the claim j=0 π r j r j! 1r+j i j r a j + h j To prove the transformation law for the functions φ r, we define for r N 0 b r := πi µ Γ 1 + r µ 8 Γ 1 + rµ! a µ r µ 0 µ r We will later show that φ r = b r The functions b r transform as described in the following proposition, a fact which follows as in [0], using Lemma 5 Proposition 6 With notation as above, for r N 0, we have that b r 1 i r b r = i r πi µ Γ 1 + r µ Γ 1 + rµ! 0 j r µ 0 µ r π r µ j 1 j r µ j! µ i j+r µ+ µ hj

12 1 KATHRIN BRINGMANN, AMANDA FOLSOM, AND ROBERT C RHOADES Our next proposition shows that the errors to modularity h r are C, a fact we use in the course of establishing the quantum modularity of the unimodal rank functions φ r in Theorem 11 In doing so, we split the Taylor expansion of Hz; into two pieces 9 Hz; = H 1 z; + H z;, with H 1 z; = H z; = h 1,r z r+1 := i h,r z r+1 := gz; ϑz; hz;, η Proposition 7 The functions h r are C on R To be more precise, h 1,r vanishes to infinite order for Q, and we extend this function to equal 0 on all of R Moreover, for H Q, the function h,r satisfies h,r = i π r+1 e πiw w r+1 sinh πw r + 1! R sinhπw dw Proof Firstly, we have that so that H 1 z; = h 1,r = i η = i η i η r z [ϑ z; h z; ] r z=0 z r+1 r r + 1 r + 1! l + 1 l=0 l=0 z r r! l+1 z l+1 [ϑ z; ] z=0 r l z r l [h z; ] z=0, r 1 l+1 l + 1!r l! z [ϑ z; ] r l l+1 z=0 z [h z; ] r l z=0 It is not hard to see that hz; is C as a function of near z = 0 Moreover by, we see that i l+1 η z [ϑ z; ] l+1 z=0 gives a linear combination of Eisenstein series multiplied by η It is well known that the Eisenstein series satisfy G k 1 = k G k k > even, G 1 = G + i 4π This implies that the function h r and its derivatives vanish exponentially for Q The second claim follows directly by inserting the Taylor expansion of e πzx

13 A FAMILY OF QUANTUM MODULAR FORMS 1 Quantum unimodal ranks Building from the results in 1 and, here we prove Theorem 11 Proof of Theorem 11 We first relate the Taylor coefficients of Y + z; to the unimodal moments u r Using the definition of u r, it is not difficult to verify that 10 U z; = u r q πizr r! Using the Taylor expansion of sinπz we find that yielding 11 Y + z; = iq 1 4 sinπzu z; = πiz πiz r a r πi = r+1 0 l r 0 l r u l qq 1 4 l r l!r l + 1! u l qq 1 4 l r l!r l + 1!, Using 11, the definition of φ r in 16, or its equivalent formulation given in 18, as well as the definition of b r in 8, it is not difficult to see that for each r N 0, b r = φ r Combining this with the fact that h j = h 1,j + h,j, Proposition 6 yields 1 φ r 1 = i r i r φ r 0 µ r 0 j r µ π r j 1 j i µ Γ 1 + r µ Γ 1 + r µ!r µ j! µ µ By continuation, 1 and what follows hold on H Q \ {0} We first consider the first summand We have by Proposition 7 µ i j+r µ+ µ h,j = i π j+1 µ µ l i j+r µ+ µ l j + 1! l l µ l 1 = l=0 R l=0 i j+r µ+ h1,j + h,j e πiw µ 1 l i µ+1 π j+1+µ l j+1 l µ 1 µ Γ j + r µ + 5 l j + 1!Γ j + r µ + 5 l i j+r+ µ l R w j+µ l+1 e πiw sinh πw coshπw dw w j+1 sinh πw coshπw dw We now define the numbers b r µ, j, l := i 1j+l+µ j+1 π r+j+µ+1 l r+l µ j 1 Γ 1 + r µ Γ j + r µ + 5 j + 1!l!µ l!r µ j!γ 1 + r Γ j + r µ + 5 l,

14 14 KATHRIN BRINGMANN, AMANDA FOLSOM, AND ROBERT C RHOADES and let 14 b r N, M := Moreover, we define 15 H r := i r 0 µ r 0 j r µ 0 µ r 0 j r µ 0 l µ N=j+µ l M=µ+l+r j b r µ, j, l π r j 1 j i µ Γ 1 + r µ Γ 1 + r µ!r µ j! µ µ i j+r µ+ h1,j Note that H r = 0 for Q \ {0} We have thus shown for H Q \ {0}, i r φ r 1 φ r = P r w, i 1 sinh πw e πiw R coshπw dw H r, as claimed in 111 Finally, under translation + 1, it is clear using the definition of φ r in 16 that φ r + 1 = e πi 1 φ r With the proof of Proposition 8 below, using 18, Theorem 11 now follows We are left to show existence of the moment function and their derivatives Proposition 8 For r, n N 0, the moment functions n [ ] q 1 n 4 ur q are defined for every root of unity q = ζ and lie in Z[ζ] Proof For ease of notation, we let D α := α α, J m w; q := wq; q m w 1 q; q m To finish the proof it is enough to show that for m sufficiently large, and every n, r N 0, the function 16 D n q D r w [J m w; q] w=1 vanishes for q = ζ It is not difficult to see that for m N D w J m w; q m 17 = J m w; q k=1 wq k 1 wq + m k k=1 w 1 q k 1 w 1 q k =: R mw; q

15 A FAMILY OF QUANTUM MODULAR FORMS 15 We further relax notation and let J := J m w; q, R := R m w; q, and R r := D r wr for r N 0 Using 17, we find that D w J = JR, D wj = J R + R 1, D wj = J R + RR 1 + R, D 4 wj = J R 4 + 4RR + R 1 + 6R R 1 + R Note that each DwJ r can be expressed as J multiplied by a sum over the partitions or r That is, given a partition π = l 1 π 1 + l π + + l r 1 π r 1 + l r π r of r where each l j π N 0 we may assign the product D j 1 w R l j π 1 j r Conversely, every such product appearing as a summand as above for DwJ r corresponds to a partition of r In general, we have that Dw r [J m w; q] w=1 = q; q m cπ D j 1 lj π w [R m w; q] w=1, π r 1 j r where we sum over all partitions π of r The exponents l j π correspond to the number of parts of the partition π of r, and the constants cπ = c r π also depend on the partition π of r Now using the definition of R m w; q in 17, we may write 18 cπ π r 1 j r D j 1 w [R m w; q] w=1 lj π = k=k1,,k c P k,r q c j=1 1 qk j r =: R m,r q, where c = c r N depends only on r, and P k,r Z[q] Next we apply the operator Dq n to q; q m multiplied by R m,r q in 18 above Using the product rule, we have that 16 equals n Dq j q; q j m D n j q R m,r q It is not difficult to see that and for l N, that 0 j n D q q; q m q; q m = m k=1 D l 1 q T m q = kq k 1 q k =: T mq, m k=1 Q k,l q 1 q k l, with Q k,l q Z[q] Therefore, we may conclude that 16 has the shape q; q m k=k1,,k d P k,r,n q d j=1 1 qk j r+n,

16 16 KATHRIN BRINGMANN, AMANDA FOLSOM, AND ROBERT C RHOADES where d = d r,n N depends only on r and n, and P k,r,n Z[q] Now if ζ = ζ m then q; q M M N vanishes at q = ζ of order m On the other hand each term M P k,r,n q d j=1 1 qk j r+n vanishes at q = ζ of order at most dr + n, which is a constant independent of m Thus, the claim follows with 4 Proof of Theorem 1 To prove Theorem 1, we recall It is not difficult to see from Proposition that Y + z; it = Hz; it + r 0 β r t r e N t β r tz r for some N > 0 To find the asymptotic expansion of Hz; it, we split as in 9 and bound using h 1,r it e M t for some M > 0 Thus we are left to determine the asymptotic expansion of H z; it For this, we write H z; it = i e πtw πwz sinh πw R cosh πw dw = i πz r+1 πt k w r+k+1 sinh πw 41 dw, r + 1! k! cosh πw k=0 where the identity in 41 refers to an asymptotic expansion Thus, to determine the asymptotic expansion of H z; it, we are left to evaluate explicitly for a N 0 w a+1 sinh πw C a := dw = 1 w a+1 e πw e πw π r 1 w a+r dw = R coshπw R coshπw r 1! R coshπw dw From 8, we have that the integral above equals i a r E a+r, yielding C a = i a 1 r=1 πi r 1 r 1! E a+r = i a 1 R r=1 πi r! r E a+r+1 The second equality above holds due to the fact that E j = 0 for j odd We are thus left to understand v r E r! r+b for positive integers b and v = πi Set fv := E r r! vr = sechv,

17 A FAMILY OF QUANTUM MODULAR FORMS 17 where the second equality above is simply the definition of the Euler numbers Then f b E r+b v = v r r! Thus 4 C a = i a 1 sech a+1 πi π = a 1 sec a+1 Next we deduce from 11 that i 1 sin πz = B 1 n πiz n 1 n! n=0 Combining the above, we have established that the asymptotic expansion of Uz; ite πt 1 as t 0 + is given by 1 πiz r 1 r 0 n r B 1 n n! 1 n r n + 1! Thus, using 4, we have the asymptotic expansion as t e πt 1 ur e πt = r! 1r r 1 k=0 t k π k k k! 0 n r Using Lemma 6 together with 4, we have that e πt 1 ur e πt = r+1 πt k r r + 1 k! n k=0 0 n r k=0 πt k C r n+k k! 1 n B 1 n n π n!r n + 1! secr n+k+1 n B n 1 5 E r+k+1 n, 6 which concludes the proof of the first statement of Theorem 1 Next we prove the claimed asymptotic for the main term Since B n+1 1 = 0, we may rewrite the k = 0 summand of 44 as 45 r+1 r n r+1 r n B n E r+1 n n 6 Now we use 6, which yields that 1 1 B n = n 1 B n 6 + a Thus, 45 equals 46 r r + 1 a=0 0 n r+1 a=0 r B n n 6 + a 5 E r+1 n 6

18 18 KATHRIN BRINGMANN, AMANDA FOLSOM, AND ROBERT C RHOADES Using 7, 46 reduces to 6 r r B r+1 + a 6 Noting again that B 1 r+1 = 0, we find that as claimed, as t 0 +, e πt 1 ur e πt 6r 5 B r+1 + B r+1 r a=0 5 An Example: the moment function φ 1 In this section, we give an exact value for the quantum moment function 51 φ 1 = 4π iq m um, n m + n q n = 4π iq 1 4 n=1 m Z u q iπ 1 u 0q To describe this, we define for positive integers n the polynomials n n d n q := nq; q n 1q n q n+ q; q n jq j 1 1 q k 5 Z[q], 5 b n q := q n+1 n j=1 q j n k=1 k j j=1 1 q k Z[q] k=1 k j Theorem 51 If h, k N, with gcdh, k = 1, we have that k k 1 h φ 1 = 8π iζ h 4k d n ζ h k k b n ζ h k n=1 Remark Theorem 51, together with 111 in the case Q \ {0} of Theorem 11, gives an exact value for the integral P 1 w, i 1 sinh πw e πiw R coshπw dw To prove Theorem 51, we first establish Proposition 5 and Proposition 5 below These propositions give alternate expressions for the functions defining φ 1 see 51, which we subsequently evaluate for q = ζ, where ζ is any root of unity Proposition 5 With notation as above, we have that u 0q = πi d n q n 1 Moreover, if gcdh, k = 1 we have that [u 0q] q=ζ h k = πi n=1 k d n ζ h k n=1

19 A FAMILY OF QUANTUM MODULAR FORMS 19 Proof The first statement follows by straightforward differentiation, using that u 0 q = 1 U0;, definition 1, and the fact that = q d To prove the second statement, πi dq we observe that d n q is of the form d n q = q; q n 1 dn q, where d n ζk h < The statement now follows, observing that for n k + 1, the factor q; q n 1 of d n q vanishes when q = ζk h Proposition 5 With notation as above, we have that πi u q = z [Uz; ] z=0 = πi b n q n 1 Moreover, if h, k N, with gcdh, k = 1, we have that k 1 πi u ζk h = πi n=1 b n ζ h k Proof of Proposition 5 The first statement follows by straightforward differentiation, using definition 1, and the fact that 1 = w d for w = πi z dw eπiz To prove the second statement, using the first statement, we see for n k, the jth summand defining b n q for any j 1 contains either the factor 1 q k or 1 q k or both, both of which vanish when q = ζk h Proof of Theorem 51 Theorem 51 now follows from the definition of φ 1 see 51, Proposition 5, and Proposition 5 References [1] M Abramowitz and I Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables, National Bureau of Standards Applied Mathematics Series, 55, Washington, DC 1964 [] C Alfes, K Bringmann, and J Lovejoy, Automorphic properties of generating functions for generalized odd rank moments and odd Durfee symbols, Math Proc Cambridge Philos Soc , [] G Andrews, Partitions with short sequences and mock theta functions, Proc Natl Acad Sci USA, , [4] G Andrews, Partitions, Durfee symbols, and the Atkin-Garvan moments of ranks, Invent Math 169 no 1 007, 7-7 [5] G Andrews, The number of smallest parts in the partitions of n, J Reine Angew Math , 1-14 [6] G Andrews and B Berndt, Ramanujan s Lost Notebook, Part II Springer, New York, 009 [7] GE Andrews, RC Rhoades, and SP Zwegers, Modularity of the concave composition generating function, Algebra and Number Theory, to appear [8] A Atkin and F Garvan, Relations between the ranks and cranks of partitions, Rankin memorial issues, Ramanujan J 7 00, 4-66 [9] A Atkin and H Swinnerton-Dyer, Some properties of partitions, Proc Lond Math Soc , [10] K Bringmann, On the explicit construction of higher deformations of partition statistics, Duke Math , 195- [11] K Bringmann and A Folsom, On the asymptotic behavior of Kac-Wakimoto characters, Proc Amer Math Soc, to appear [1] K Bringmann, F Garvan, and K Mahlburg, Partition statistics and quasiharmonic Maass forms, Int Math Res Not 009, 6-97 [1] K Bringmann, J Lovejoy, and R Osburn, Automorphic properties of generating functions for generalized rank moments and Durfee symbols, Int Math Res Not 010, no, 8-60

20 0 KATHRIN BRINGMANN, AMANDA FOLSOM, AND ROBERT C RHOADES [14] K Bringmann, K Mahlburg, and R Rhoades, Peak positions of strongly unimodal sequences, in preparation [15] K Bringmann and K Ono, Dyson s ranks and Maass forms, Ann of Math , [16] J Bryson, K Ono, S Pitman, and R C Rhoades, Unimodal sequences and quantum and mock modular forms, Proc Natl Acad Sci USA , no 40, [17] D Cvijovic, Closed-form formulae for the derivatives of trigonometric functions at rational multiples of π, Appl Math Lett 009, [18] A Dabholkar, S Murthy, and D Zagier Quantum black holes, wall crossing, and mock modular forms, submitted for publication [19] F Dyson, Some guesses in the theory of partitions, Eureka Cambridge , [0] M Eichler and D Zagier, The theory of Jacobi forms, Progr Math, 55 Birkhäuser Boston, Boston, MA, 1985 [1] A Erdelyi, W Magnus, F Oberhettinger, and F Tricomi, Higher transcendental functions Vol I Based, in part, on notes left by Harry Bateman and compiled by the Staff of the Bateman Manuscript Project, reprint of the 1955 original, Robert E Krieger Publishing, Malabar 1981 [] F Garvan, Higher order spt-functions, Adv Math 8 011, [] V Kac and M Wakimoto, Integrable highest weight modules over affine superalgebras and Appell s function, Comm Math Phys , [4] J Manschot, Wall-crossing of D4-branes using flow trees, Adv Theor Math Phys [5] A Mellit and S Okada, Joyce invariants for K surfaces and mock theta functions, Comm Number Theory Phys 009, [6] H Rademacher, Topics in analytic number theory, Die Grundlehren der math Wiss, Band 169, Springer- Verlag, Berlin, 197 [7] S Ramanujan, Some properties of pn, number of partitions of n, Proc Cambridge Phil Soc XIX 1919, [8] G Shimura, On modular forms of half integral weight, Ann of Math , [9] C Vafa and E Witten, A strong coupling test of S-duality, Nuclear Phys B , -77 [0] D Zagier, Periods of modular forms and Jacobi theta functions, Invent Math , [1] D Zagier, Vassiliev invariants and a strange identity related to the Dedekind eta-function, Topology , [] D Zagier, Quantum modular forms, Quanta of maths, , Clay Math Proc, 11, Amer Math Soc, Providence, RI, 010 [] S Zwegers, Mock theta functions, PhD Thesis, Universiteit Utrecht, 00 [4] S Zwegers, Multivariable Appell functions, preprint Mathematical Institute, University of Cologne, 5091 Cologne, Germany address: kbringma@mathuni-koelnde Yale University, Mathematics Department, PO Box 088 New Haven, CT address: amandafolsom@yaleedu Center for Communications Research, 805 Bunn Dr, Princeton, NJ address: robrhoades@gmailcom