Rank–Crank-type PDEs and generalized Lambert series identities

RANK-CRANK TYPE PDES AND GENERALIZED LAMBERT SERIES IDENTITIES arXiv:1201.1663v1 [math.NT] 8 Jan 2012 SONG HENG CHAN, ATUL DIXIT AND FRANK G. GARVAN Dedicated to our friends, Mourad Ismail and Dennis Stanton Abstract. We show how Rank-Crank type PDEs for higher order Appell functions due to Zwegers may be obtained from a generalized Lambert series identity due to the first author. Special cases are the Rank-Crank PDE due to Atkin and the third author and a PDE for a level 5 Appell function also found by the third author. These two special PDEs are related to generalized Lambert series identities due to Watson, and Jackson respectively. The first author’s Lambert series identities are common generalizations. We also show how Atkin and Swinnerton-Dyer’s proof using elliptic functions can be extended to prove these generalized Lambert series identities. 1. Introduction F. J. Dyson [9], [10, p. 52] defined the rank of a partition as the largest part minus the number of parts. Dyson conjectured that the residue of the rank mod 5 divides the partitions of 5n + 4 into 5 equal clases thereby providing a combinatorial interpretation of Ramanujan’s famous partition congruences p(5n + 4) ≡ 0 (mod 5). He also conjectured that the rank mod 7 likewise gives Ramanujan’s partition congruence p(7n + 5) ≡ 0 (mod 7). Dyson’s rank conjectures were proved by A. O. L. Atkin and H. P. F. Swinnerton-Dyer [3]. The following was the crucial identity that Atkin and Swinnerton-Dyer needed for the proof of the Dyson rank conjectures. It was first proved by G.N. Watson [18]. ∞ [ζ 2 ]∞ X (−1)n q 3n(n+1)/2 [ζ]∞ [ζ 2]∞ (q)2∞ ζ + [ζ]∞ n=−∞ 1 − zq n [z/ζ]∞ [z]∞ [ζz]∞   ∞ X ζ −3n ζ 3n+3 n 3n(n+1)/2 = (−1) q . + n /ζ n 1 − zq 1 − zζq n=−∞ (1.1) Date: January 8, 2012. 2010 Mathematics Subject Classification. 11F11, 11P82, 11P83, 33D15. Key words and phrases. Rank-Crank PDE, higher level Appell function, Lambert series, partition, q-series, basic hypergeometric function, quasimodular form. The third author was supported in part by NSA Grant H98230-09-1-0051. 1 2 SONG HENG CHAN, ATUL DIXIT AND FRANK G. GARVAN Throughout we use the standard q-notation (x)0 := (x; q)0 := 1, (x)n := (x; q)n := n−1 Y m=0 (1 − xq m ), (x1 , · · · , xm )n := (x1 , · · · , xm ; q)n := (x1 ; q)n · · · (xm ; q)n , [x1 , · · · , xm ]n := [x1 , · · · , xm ; q]n := (x1 , q/x1 , · · · , xm , q/xm ; q)n . when n is a nonnegative integer. Assuming |q| < 1 we also use this notation when n = ∞ by interpreting its meaning as the limit as n → ∞. Later M. Jackson [14] proved an analogue of the above identity, ∞ [ζ]∞ [ζ 2 ]∞ [xζ]∞ [ζ/x]∞ (q)2∞ ζ 2[ζ 2 ]∞ [xζ]∞ [x/ζ]∞ X (−1)n q 5n(n+1)/2 + [ζ]∞ [x]2∞ 1 − zq n [z/x]∞ [z/ζ]∞ [z]∞ [zζ]∞ [zx]∞ n=−∞   ∞ ζ [ζ]∞ [ζ 2]∞ X x−5n x5n+5 n 5n(n+1)/2 + (−1) q (1.2) + x [x]∞ [x2 ]∞ n=−∞ 1 − zq n /x 1 − zxq n   ∞ X ζ 5n+5 ζ −5n n 5n(n+1)/2 . + = (−1) q −1 q n n 1 − zζ 1 − zζq n=−∞ Recently, the first author [8, p.603] found a generalization of the above two identities, namely, ∞ xm [x2 /x1 , · · · , xm /x1 , x1 xm , · · · , x1 x2 , x2 ]∞ X (−1)n q (2m+1)n(n+1)/2 1 1 [x1 ]∞ [x2 , · · · , xm ]2∞ n=−∞ 2 , x1 x2 , x1 ]∞ (q)2∞ 1 − zq n [x1 /x2 , · · · , x1 /xm , x1 , x1 xm , · · · [z/x1 , z/x2 , · · · , z/xm , z, zxm , · · · , zx1 ]∞  x1 [x1 /x3 , · · · , x1 /xm , x1 , x1 xm , · · · , x1 x3 , x21 ]∞ + (1.3) x2 [x2 /x3 , · · · , x2 /xm , x2 , x2 xm , · · · , x22 ]∞ !  ∞ (2m+1)(n+1) −(2m+1)n X x x 2 2 n (2m+1)n(n+1)/2 × (−1) q + + idem(x2 ; x3 , · · · , xm ) 1 − zq n /x2 1 − zx2 q n n=−∞ ! ∞ −(2m+1)n (2m+1)(n+1) X x x 1 (−1)n q (2m+1)n(n+1)/2 = + 1 , n /x n 1 − zq 1 − zx q 1 1 n=−∞ + where g(a1 , a2 , · · · , am )+ idem(a1 ; a2 , · · · , an ) denotes the sum n X i=1 g(ai , a2 , · · · , ai−1 , a1 , ai+1 , · · · , am ), in which the i-th term of the sum is obtained from the first by interchanging a1 and ai . Equation (1.3) was proved using partial fractions. Indeed, the m = 1 case of (1.3) is equivalent to (1.1), while the m = 2 case is equivalent to (1.2). The fact that the right-hand side of (1.2) is independent of x, and that the right-hand side of (1.3) is RANK-CRANK TYPE PDES 3 independent of x2 , x3 , · · · , xm seems to be intriguing at first. Indeed, one purpose of this article is to show that the left-hand sides of (1.2) and (1.3) are really elliptic functions of order less than 2, in fact entire functions as we show, in the respective variables (x for (1.2) and x2 for (1.3) while holding x3 , · · · , xm fixed) and therefore that they must be constants which are nothing but the right-hand sides of (1.2) and (1.3) respectively. Since (1.2) follows from (1.3), we show this only for (1.3). This is done in Section 2. Let N(m, n) denote the number of partitions of n with rank m. Then the rank generating function R(z, q) is given by 2 ∞ ∞ X n X X qn m n N(m, n)z q = R(z, q) = . (1.4) (zq)n (z −1 q)n n=0 n=0 m=−n In [1], G. E. Andrews and the third author defined the crank of a partition, a partition statistic hypothesized by Dyson in [9]. It is the largest part if the partition contains no ones, and otherwise is the number of parts larger than the number of ones minus the number of ones. For n > 1, we let M(m, n) denote the number of partitions of n with crank m. If we amend the definition of M(m, n) for n = 1, then the generating function can be given as an infinite product. Accordingly, we assume M(0, 1) = −1, M(−1, 1) = M(1, 1) = 1, and M(m, 1) = 0 otherwise. Then the crank generating function C(z, q) is given by ∞ X n X (q)∞ . C(z, q) = M(m, n)z m q n = −1 q) (zq) (z ∞ ∞ n=0 m=−n (1.5) Atkin and the third author [2] found the so-called Rank-Crank PDE, a partial differential equation (PDE) which relates R(z, q) and C(z, q). To state this PDE in its original form, we first define the differential operators ∂ ∂ (1.6) δz = z , δq = q . ∂z ∂q Then the Rank-Crank PDE can be written as   1 2 1 2 ∗ 3 (1.7) z(q)∞ [C (z, q)] = 3δq + δz + δz R∗ (z, q), 2 2 where R(z, q) R∗ (z, q) := , 1−z C(z, q) C ∗ (z, q) := . (1.8) 1−z In [2], it was shown how the Rank-Crank PDE and certain results for the derivatives of Eisenstein series lead to exact relations between rank and crank moments. As in [13], define Nk (m, n) by ∞ X 1 X n Nk (m, n)q = (−1)n−1 q n((2k−1)n−1)/2+|m|n (1 − q n ), (1.9) (q) ∞ n=1 n≥0 4 SONG HENG CHAN, ATUL DIXIT AND FRANK G. GARVAN of any positive integer k. When k = 1 this is the generating function for the crank, and when k = 2 it is the generating function for the rank. When k ≥ 2, Nk (m, n) can be interpreted combinatorially as the number of partitions of n into k −1 successive Durfee squares with k-rank equal to m. See [13, Eq.(1.11)] for a definition of the k-rank. We define n X X Rk (z, q) := Nk (m, n)z m q n . (1.10) n≥0 m=−n From [13, Eq.(4.5)], this generating function can be written as 2 Rk (z, q) = X nk−1 ≥nk−2 ≥···≥n1 2 2 q n1 +n2 +···+nk−1 , (q)nk−1 −nk−2 · · · (q)n2 −n1 (zq)n1 (z −1 q)n1 ≥1 (1.11) when k ≥ 2. In Section 3, we show that Rk (z, q) is related to the level 2k − 1 Appell function ∞ X (−1)n q (2k−1)n(n+1)/2 (2k−1) Σ (z, q) := . (1.12) n 1 − zq n=−∞ We obtain the following Theorem 1.1. For k ≥ 1, Rk (z, q) 1 = (q)∞ z k−1 (1 − z)Σ(2k−1) (z, q) − zθ1,2k−1 (q) + z(1 − z) where θj,2k−1(q) = ∞ X k−3 X m=0 (−1)n q n((2k−1)n+j)/2 , ! z m θ2m+3,2k−1 (q) , (1.13) (1.14) n=−∞ for j = 1, 3, . . . , 2k − 3. This theorem generalizes Lemma 7.9 in [12] which gives a relation between the rank generating function R(z, q) and a level 3 Appell function. The k = 1 case of the theorem gives the familiar partial fraction expansion of Jacobi’s theta product Pfor the reciprocal n n(n+1)/2 (z)∞ (z −1 q)∞ [16, p. 1], [17, p. 136], since ∞ (−1) q = 0. n=−∞ th A few years ago the third author found a 4 order PDE, which is an analogue of the Rank-Crank PDE and is related to the 3-rank [13]. To state this PDE we define 1 G(5) (z, q) := Σ(5) (z, q). (1.15) (q)3∞ Then 24(q)2∞ [C ∗ (z, q)]5 = 24(1 − 10Φ3 (q))G(5) (z, q)
+ 100δq + 50δz + 100δq δz + 35δz2 + 20δq δz2 + 100δq2 + 10δz3 + δz4 G(5) (z, q), (1.16) RANK-CRANK TYPE PDES 5 where ∞ X n3 q n Φ3 (q) := . 1 − qn n=1 (1.17) This PDE can be written more compactly as 24(q)2∞ [C ∗ (z, q)]5 = (H2∗ − E4 ) G(5) (z, q), where H∗ is the operator (1.18) H∗ := 5 + 10δq + 5δz + δz2 , and E4 := E4 (q) := 1 + 240Φ3 (q), is the usual Eisenstein series of weight 4. The PDE (1.16) was first conjectured by the third author and then subsequently proved and generalized by Zwegers [21]. It was also Zwegers who first observed that (1.16) could be written in a more compact form. We now describe Zwegers’s generalization. Define for l ∈ Z>0 , the level l Appell function as ∞ X (−1)ln q ln(n+1)/2 w n l/2 , (1.19) Al (u, v) := Al (u, v; τ ) := z 1 − zq n n=−∞ where z = e2πiu , w = e2πiv , q = e2πiτ , and define the modified rank and crank generating functions as follows. z 1/2 q −1/24 R(z, q), 1−z z 1/2 q −1/24 C := C(u; τ ) := C(z, q). 1−z R := R(u; τ ) := (1.20) (1.21) Here and throughout we assume Im τ > 0 so that |q| < 1. Then the following theorem due to Zwegers gives for odd l, the (l − 1)th order analogue of the Rank-Crank PDE. Theorem 1.2 (Zwegers[21]). Let l ≥ 3 be an odd integer. Define Hk := 1 l(2k − 1) l ∂ ∂2 + − E2 , 2 2 πi ∂τ (2πi) ∂u 12 H[k] := H2k−1 H2k−3 · · · H3 H1 , (1.22) P∞ where E2P (τ ) = 1 − 24 n=1 σ1 (n)q n is the usual Eisenstein series of weight 2 with σα (n) = d|n dα . Then there exist holomorphic modular forms fj (which can be constructed explicitly), with j = 4, 6, 8, · · · , l − 1, on SL2 (Z) of weight j, such that   (l−5)/2 X H[(l−1)/2] + fl−2k−1 H[k]  Al (u, 0) = (l − 1)!η l C l , (1.23) k=0 where η is the Dedekind η-function, given by η(τ ) = q 1/24 (q)∞ . 6 SONG HENG CHAN, ATUL DIXIT AND FRANK G. GARVAN Zwegers proved this theorem using the formulas and methods motivated from the theory of Jacobi forms. In contrast to this, the proof of the Rank-Crank PDE by Atkin and the third author, which corresponds to the l = 3 case of Zwegers’s PDE, depends upon simply taking the second derivative with respect to ζ of both sides of (1.1). The main goal of this paper is to show how a generalized Rank-Crank PDE of any odd order follows from the Lambert series identity (1.3) in a similar fashion. We will obtain Zwegers’s result in a different form. In our form the coefficients are quasimodular forms rather than holomorphic modular forms, but in contrast, our coefficients are given recursively. See Theorem 4.4 and Corollary 4.5. This paper is organized as follows. In Section 2, we prove (1.3) using the theory of elliptic functions. Then in Section 3 we prove Theorem 1.1, which is the theorem that relates Rk (z, q) with the level 2k − 1 Appell function Σ(2k−1) (z, q). In Section 4 we prove our main result that shows how (1.3) can be used to derive the higher order Rank-Crank-type PDEs of Zwegers. In the light of (1.19), it should be observed that the identities (1.1) and (1.2) are really the identities involving certain combinations of level 3 and level 5 Appell functions respectively while (1.3) is an identity involving a combination of level (2m + 1) Appell functions. However, the analogue for level 1 Appell function which cannot be derived from (1.3) was found by R. Lewis [15, Equation 11] and is as follows.   ∞ X [z]∞ [ζ 2 ]∞ (q)2∞ ζ −n ζ n+1 n n(n+1)/2 (−1) q = + . (1.24) [zζ]∞ [ζ]∞ [zζ −1 ]∞ n=−∞ 1 − zq n /ζ 1 − zζq n 2. General Lambert series identity through elliptic function theory Atkin and Swinnerton-Dyer’s proof of (1.1) depends in essence on the theory of elliptic functions. In this section, we show how this method of proof can be used to prove (1.3). Let z = e2πiu , x1 = e2πiv , x2 = e2πiw , (2.1) and let xj = e2πiaj , j = 3, · · · , m, (2.2) where u, v, w, a3, . . . , am are all complex numbers. Also recall that q = e2πiτ , where Im τ > 0. Let [b]∞ =: J(a, q) =: J(a), where b = e2πia , a ∈ C. (2.3) Then using the Jacobi triple product identity [4, p. 10, Theorem 1.3.3], we easily find that, ieπia q −1/8 J(a, q) = θ(a), (2.4) (q)∞ where      ∞ X 1 1 1 2 τ +2πi n+ z+ πi n+ 2 2 2 . (2.5) θ(z) = θ(z; τ ) := e n=−∞ RANK-CRANK TYPE PDES 7 Comparing this with the classical definition of θ1 (z) [11, p. 355, Section 13.19, Equation 10], we find that upon replacing q by q 1/2 in this classical definition, θ(z) = −θ1 (z). From [20, p. 8], we see that θ(z + 1) = −θ(z), θ(z + τ ) = −e−πiτ −2πiz θ(z), θ(−z) = −θ(z), ′ θ (0; τ ) = −2πq 1/8 (q)3∞ . (2.6) (2.7) (2.8) (2.9) Using (2.6) and (2.7), we have J(a + 1, q) = J(a, q), (2.10) J(a + τ, q) = −e−2πia J(a, q), (2.11) J(−a, q) = −e−2πia J(a, q). (2.13) J(a − τ, q) = −q −1 e2πia J(a, q), (2.12) Using (2.3), we rephrase (1.3) as follows: ∞ e2πimv J(w − v)J(w + v)J(2v) Y J(aj − v)J(aj + v) X (−1)n q (2m+1)n(n+1)/2 J(v)(J(w))2 (J(aj ))2 1 − e2πiu q n n=−∞ 3≤j≤m Y J(v − aj )J(v + aj ) J(v − w)J(v)J(v + w)J(2v)(q)2∞ J(u − v)J(u − w)J(u)J(u + w)J(u + v) 3≤j≤m J(u − aj )J(u + aj )  J(v)J(2v) Y J(v − aj )J(v + aj ) + e2πi(v−w) J(w)J(2w) 3≤j≤m J(w − aj )J(w + aj )  −2πiw(2m+1)n   ∞ X e e2πiw(2m+1)(n+1) n (2m+1)n(n+1)/2 × (−1) q + + idem(w; a3, · · · , am ) 1 − e2πi(u−w) q n 1 − e2πi(u+w) q n n=−∞  −2πiv(2m+1)n  ∞ X e e2πiv(2m+1)(n+1) n (2m+1)n(n+1)/2 (−1) q = + . (2.14) 1 − e2πi(u−v) q n 1 − e2πi(u+v) q n n=−∞ + Fix a3 , · · · , am , consider the left-hand side of (2.14) as a function of w only and denote it by g(w). Let f1 (w) denote the expression in line 1 of (2.14), f2 (w) the expression in line 2 of (2.14) and f3 (w) the expression in lines 3 and 4 of (2.14). Then, using (2.10), (2.11) and (2.12), we see f1 (w + 1) = f1 (w) = f1 (w + τ ), f2 (w + 1) = f2 (w) = f2 (w + τ ), f3 (w + 1) = f3 (w). (2.15) 8 SONG HENG CHAN, ATUL DIXIT AND FRANK G. GARVAN Another application of (2.11) and (2.12) gives f3 (w + τ ) J(v − aj )J(v + aj ) e2πi(v−w−τ ) J(v)J(2v) Y = J(w + τ )J(2w + 2τ ) 3≤j≤m J(w + τ − aj )J(w + τ + aj )  −2πi(w+τ )(2m+1)n  ∞ X e e2πi(w+τ )(2m+1)(n+1) n (2m+1)n(n+1)/2 × (−1) q + 1 − e2πi(u−w−τ ) q n 1 − e2πi(u+w+τ ) q n n=−∞ + m X e2πi(v−ak ) k=3 ∞ X J(v)J(2v)J(v − w − τ )J(v + w + τ ) J(ak )J(2ak )J(ak − w − τ )J(ak + w + τ ) Y 2<j<m+1 j6=k J(v − aj )J(v + aj ) J(ak − aj )J(ak + aj )  e−2πiak (2m+1)n e2πiak (2m+1)(n+1) + (−1) q × 2πi(u−ak ) q n 1 − e 1 − e2πi(u+ak ) q n n=−∞ ∞ Y J(v − aj )J(v + aj )  X 2πiv+4πimw J(v)J(2v) =e (−1)n q (2m+1)(n+1)(n+2)/2 J(w)J(2w) 3≤j≤m J(w − aj )J(w + aj ) n=−∞  ∞ 2πiw(2m+1)n (2m+1)n X q e−2πiw(2m+1)(n+1) q −(2m+1)(n+1) n (2m+1)n(n−1)/2 e (−1) q + × 1 − e2πi(u−w) q n 1 − e2πi(u+w) q n n=−∞ + m X n (2m+1)n(n+1)/2 e2πi(v−ak ) k=3 × = f3 (w). ∞ X J(v)J(2v)J(v − w)J(v + w) J(ak )J(2ak )J(ak − w)J(ak + w) n (2m+1)n(n+1)/2 (−1) q n=−∞   Y 2<j<m+1 j6=k J(v − aj )J(v + aj ) J(ak − aj )J(ak + aj ) e2πiak (2m+1)(n+1) e−2πiak (2m+1)n + 1 − e2πi(u−ak ) q n 1 − e2πi(u+ak ) q n  (2.16) Thus from (2.15) and (2.16), we deduce that g is a doubly periodic function in w with periods 1 and τ . Our next task is to show that g is an entire function of w and hence a constant (with respect to w). We show that the poles of g at w = u and w = −u are actually removable singularities by proving that limw→±u (w ∓ u) (f2 (w) + f3 (w)) = 0 which readily implies that limw→±u (w ∓ u)g(w) = 0. Let Y A := A(v, a3 , · · · , am ; q) := J(v)J(2v) J(v − aj )J(v + aj ). (2.17) 3≤j≤m Next, applying (2.4), (2.13) and (2.9), we see that lim (w − u) (f2 (w) + f3 (w))  Y 1 J(v − w)J(v + w)(q)2∞ = A lim (w − u) w→u J(u − w)J(u + w)J(u − v)J(u)J(u + v) 3≤j≤m J(u − aj )J(u + aj )  ∞ X 1 1 e2πi(v−w) Y + (−1)n q (2m+1)n(n+1)/2 + 2πi(u−w) J(w)J(2w) 3≤j≤m J(w − aj )J(w + aj ) 1 − e n=−∞ w→u n6=0 RANK-CRANK TYPE PDES 9  ∞ 2πiw(2m+1)(n+1) X e−2πiw(2m+1)n n (2m+1)n(n+1)/2 e + (−1) q × 1 − e2πi(u−w) q n n=−∞ 1 − e2πi(u+w) q n   e2πi(v−u) Y 1 −iq 1/8 (q)3∞ 1 =A + J(u)J(2u) 3≤j≤m J(u − aj )J(u + aj ) θ′ (0) 2πi = 0. (2.18) Similarly, limw→−u (w +u) (f2 (w) + f3 (w)) = 0. Now the only other possibility of a pole of g is at 0, which arises from f1 and f3 each having a pole at 0. Again, to show that this is a removable singularity, it suffices to show that limw→0 w (f1 (w) + f3 (w)) = 0. To show this, we need Jacobi’s duplication formula for theta functions [19, p. 488, Ex. 5] θ(2w)θ2 θ3 θ4 = 2θ(w)θ2 (w)θ3 (w)θ4 (w). (2.19) Let ∞ Y J(aj − v)J(aj + v) X (−1)n q (2m+1)n(n+1)/2 . J(v) 3≤j≤m (J(aj ))2 1 − e2πiu q n n=−∞ 2πimv J(2v) B := B(u, v, a3, · · · , am ; q) := e (2.20) Then from (2.14) and (2.20), lim w (f1 (w) + f3 (w))  1 J(w − v)J(w + v) e2πi(v−w) Y = lim w B + A w→0 (J(w))2 J(w)J(2w) 3≤j≤m J(w − aj )J(w + aj )   ∞ X e2πiw(2m+1)(n+1) e−2πiw(2m+1)n n (2m+1)n(n+1)/2 + × (−1) q 1 − e2πi(u−w) q n 1 − e2πi(u+w) q n n=−∞ w→0 + lim w w→0 m X e2πi(v−ak ) k=3 J(v)J(2v)J(v − w)J(v + w) J(ak )J(2ak )J(ak − w)J(ak + w) Y 2<j<m+1 j6=k J(v − aj )J(v + aj ) J(ak − aj )J(ak + aj ) −5πiw Y w 1 θ(w − v)θ(w + v) 2πiv 1/4 2 e = lim B − Ae q (q)∞ w→0 θ(w) θ(w) θ(2w) 3≤j≤m J(w − aj )J(w + aj )   −2πiw(2m+1)n ∞ X e2πiw(2m+1)(n+1) e n (2m+1)n(n+1)/2 + × (−1) q 1 − e2πi(u−w) q n 1 − e2πi(u+w) q n n=−∞  = 1 D(w) lim , θ (0) w→0 θ(w) ′ (2.21) 10 SONG HENG CHAN, ATUL DIXIT AND FRANK G. GARVAN where D(w) := D(w, v, a3, · · · , am ; q) := Bθ(w − v)θ(w + v) − Ae2πiv q 1/4 (q)2∞ E(w) 1 e−5πiw θ(w) Y E(w) := E(w; u, a3 · · · , am ; q) := θ(2w) 3≤j≤m J(w − aj )J(w + aj )   −2πiw(2m+1)n ∞ X e2πiw(2m+1)(n+1) e n (2m+1)n(n+1)/2 . + × (−1) q 2πi(u−w) q n 2πi(u+w) q n 1 − e 1 − e n=−∞ (2.22) Now using (2.19), (2.4) and (2.13), we find that as w → 0, ∞ X (−1)n q (2m+1)n(n+1)/2 e2πiv Aq 1/4 (q)2∞ D(w) → −Bθ (v) − (−1)m−2 e−2πi(a3 +···+am ) (J(a3 ) · · · J(am ))2 n=−∞ 1 − e2πiu q n 2 = 0, (2.23) which is observed by putting the expressions for B and C back in the first expression on the right side in (2.23). Thus, ′ D (0) lim w (f1 (w) + f3 (w)) = ′ 2 . w→0 θ (0) (2.24) ′ Now we calculate D (0).  ′  ′ ′ ′ D (w) = B θ (w − v)θ(w + v) + θ(w − v)θ (w + v) − e2πiv Aq 1/4 (q)2∞ E (w). (2.25) Using (2.4) and (2.19), we have 1 e−πiw(2m+1) θ2 θ3 θ4 Y 2θ2 (w)θ3 (w)θ4 (w) 3≤j≤m θ(w − aj )θ(w + aj )   −2πiw(2m+1)n ∞ X e2πiw(2m+1)(n+1) e n (2m+1)n(n+1)/2 . + × (−1) q 2πi(u−w) q n 2πi(u+w) q n 1 − e 1 − e n=−∞ E(w) = (−1)m−2 q m−2 4 2(m−2) (q)∞ (2.26) Differentiating both sides with respect to w and simplifying, we obtain m−2 1 ′ 2(m−2) −πiw(2m+1) E (w) = (−1)m−2 q 4 (q)∞ e θ2 θ3 θ4 2 ′ ′ ′ X  θ′ (w − aj ) θ′ (w + aj )  θ2 (w) θ3 (w) θ3 (w) +  πi(2m + 1) + θ2 (w) + θ3 (w) + θ3 (w) + θ(w − aj ) θ(w + aj ) 3≤j≤m Y × − θ2 (w)θ3 (w)θ4 (w) θ(w − aj )θ(w + aj ) 3≤j≤m ∞ X e2πiw(2m+1)(n+1) e + × (−1) q 1 − e2πi(u−w) q n 1 − e2πi(u+w) q n n=−∞  Y 1 1 ′ + F (w) , θ2 (w)θ3 (w)θ4 (w) 3≤j≤m θ(w − aj )θ(w + aj ) n (2m+1)n(n+1)/2  −2πiw(2m+1)n  (2.27) RANK-CRANK TYPE PDES 11 where F (w) := F (w, u, m; q) := ∞ X n (2m+1)n(n+1)/2 (−1) q n=−∞   e2πiw(2m+1)(n+1) e−2πiw(2m+1)n . + 1 − e2πi(u−w) q n 1 − e2πi(u+w) q n (2.28) It is straightforward to see that ′ F (0) = 2πi(2m + 1) From (2.8), we have ∞ X (−1)n q (2m+1)n(n+1)/2 . 2πiu q n 1 − e n=−∞ ′ (2.29) ′ θ (−z) = θ (z). (2.30) ′ Then letting w → 0 in (2.27), and using (2.8), (2.30), (2.29) and the fact that θk (0) = 0 for 2 ≤ k ≤ 4, we find that ′ E (0) = 0. (2.31) ′ Using (2.30) and (2.31) in (2.25), we finally deduce that D (0) = 0. With the help of (2.24), this then implies that limw→0 w (f1 (w) + f3 (w)) = 0 and thus limw→0 wg(w) = 0. Thus w = 0 is also a removable singularity, which implies that g is an doubly periodic entire function and hence a constant, say K (which may very well depend on v). Finally, since J(0) = 0, we have   −2πiv(2m+1)n ∞ X e2πiv(2m+1)(n+1) e n (2m+1)n(n+1)/2 . + K = g(v) = (−1) q 1 − e2πi(u−v) q n 1 − e2πi(u+v) q n n=−∞ This completes the proof. 3. Proof of Theorem 1.1 From [13, Eq.(4.3)], we see that   ∞ 1 X 1 z −1 q n n−1 n((2k−1)n−1)/2 n Rk (z, q) = (−1) q (1 − q ) + (q)∞ n=1 1 − zq n 1 − z −1 q n ∞ z −1 X 1 − qn = (−1)n−1 q n((2k−1)n+1)/2 . (q)∞ n=−∞ 1 − z −1 q n n6=0 (3.1) Replacing z by z −1 in (1.11) and (3.1), we see that Rk (z, q) ∞ z X 1 − qn = (−1)n−1 q n((2k−1)n+1)/2 (q)∞ n=−∞ 1 − zq n n6=0   ∞ (1 − z)q n z X n−1 n((2k−1)n+1)/2 1− (−1) q = (q)∞ n=−∞ 1 − zq n n6=0 12 = SONG HENG CHAN, ATUL DIXIT AND FRANK G. GARVAN  ∞ X (−1)n q n((2k−1)n+3)/2  z  (−1)n−1 q n((2k−1)n+1)/2 + (1 − z)   (q)∞ n=−∞ 1 − zq n n=−∞ n6=0 =  ∞ X  n6=0 ∞ X ∞ X  z  (−1)n q n((2k−1)n+3)/2  (−1)n q n((2k−1)n+1)/2 + (1 − z) 1 −  (q)∞ 1 − zq n n=−∞ n=−∞ n6=0   ∞ X z −zθ1,2k−1 (q) (−1)n q n((2k−1)n+3)/2 + 1 + (1 − z) = (q)∞ (q)∞ 1 − zq n n=−∞ n6=0   k−2 (k−2)n  ∞ X z q z 1 − (zq n )k−2 −zθ1,2k−1 (q) n n((2k−1)n+3)/2 1 + (1 − z) (−1) q + + = (q)∞ (q)∞ 1 − zq n 1 − zq n n=−∞ n6=0  ∞ X −zθ1,2k−1 (q) z = + 1 + (1 − z) (−1)n q n((2k−1)n+3)/2 (q)∞ (q)∞ n=−∞ z k−2 (k−2)n n6=0 q 1 − zq n + k−3 X z m q mn m=0 !  ∞ X z (−1)n q (2k−1)n(n+1)/2 −zθ1,2k−1 (q) k−2 1 + z (1 − z) + = (q)∞ (q)∞ 1 − zq n n=−∞ n6=0 + (1 − z) k−3 X z m m=0 ∞ X n n((2k−1)n+2m+3)/2 (−1) q n=−∞ n6=0   ∞ X z (−1)n q (2k−1)n(n+1)/2 −zθ1,2k−1 (q) z k−2 (1 − z) + = (q)∞ (q)∞ 1 − zq n n=−∞ + (1 − z) 1 = (q)∞ k−3 X z m m=0 ∞ X n n((2k−1)n+2m+3)/2 (−1) q n=−∞ −zθ1,2k−1 (q) + z k−1 (1 − z)Σ(k) (z, q) + z(1 − z) This completes the proof of Theorem 1.1. k−3 X  ! z m θ2m+3,2k−1 (q) . m=0 4. Higher order Rank-Crank-type PDEs In this section we show how the generalized Lambert series identity (1.3) can be used to derive general Rank-Crank PDEs of the type found by Zwegers. 4.1. The idea. First we let xi = ζ i , 1 ≤ i ≤ m in (1.3) to obtain Ym (ζ, z, q) (q)2∞ = S2m+1 (ζ, z, q)+ m−1 X Fj,m (ζ, q) S2m+1(ζ j+1, z, q)−F0,m (ζ, q) Σ(2m+1) (z, q), j=1 (4.1) RANK-CRANK TYPE PDES where Sk (ζ, z, q) := ∞ X n kn(n+1)/2 (−1) q n=−∞ for k odd and  13 ζ −kn ζ k(n+1) + 1 − zζ −1 q n 1 − zζq n  , (4.2) [ζ m+1 ] F0,m (ζ, q) := ζ m m ∞ , [ζ ]  −(m−1)∞ −(m−2)  ζ ,ζ , · · · , ζ −(m−j) ∞ Fj,m (ζ, q) := (for 1 ≤ j ≤ m − 1), [ζ m+2 , · · · , ζ m+j+1]∞  −(m−1) −(m−2)  ζ ,ζ , · · · , ζ −2, ζ −1, ζ, ζ 2, · · · , ζ m, ζ m+1 ∞ . Ym (ζ, z, q) := [zζ −m , zζ −(m−1) , · · · , zζ −2 , zζ −1 , z, zζ, zζ 2 , · · · , zζ m−1 , zζ m ]∞ (4.3) (4.4) (4.5) The basic idea is to apply the operator D2m to both sides of (4.1) where  ℓ ∂ Dℓ := ζ = δζℓ ζ=1 . ∂ζ (4.6) ζ=1 We will also need the differential operator Hk∗ := kδz + 2kδq + δz2 . (4.7) We note that the operator Hk∗ differs from Zwegers’s Hk although they are similar. First we need to write the functions Σ(k) (z, q) and Sk (z, q) as double series. Throughout we assume that 0 < |q| < 1, z 6∈ {q n : n ∈ Z} ∪ {0} and ζ 6∈ {z ±1 q n : n ∈ Z} ∪ {0}. We obtain ! ∞ ∞ ∞ X X X z m ζ m q mn z m ζ −mq mn + ζ k(n+1) Sk (ζ, z, q) = (−1)n q kn(n+1)/2 ζ −kn − = n=0 ∞ X (−1)n q kn(n−1)/2 n=1 ∞ X ∞ X ζ kn m=0 ∞ X z −m ζ m q mn + ζ k(−n+1) − and (k) (−1)n z m q kn(n+1)/2+mn ζ −kn−m + ζ k(n+1)+m Σ (z, q) = n m kn(n+1)/2+mn (−1) z q n=0 m=0 We have (−1)n z −m q kn(n−1)/2+mn ζ kn+m + ζ −kn+k−m − ∞ X ∞ X !

(4.8) (−1)n z −m q kn(n−1)/2+mn . (4.9) n=1 m=1 ∞ X ∞ X z −m ζ −mq mn m=1 m=1 n=0 m=0 ∞ ∞ X X m=0 ∞ X n=1 m=1 Theorem 4.1. Suppose k is odd and 1 ≤ ℓ ≤ k − 1. Then Dℓ Sk (ζ, z, q) = Pk,ℓ(Hk∗ )Σ(k) (z, q), (4.10) 14 SONG HENG CHAN, ATUL DIXIT AND FRANK G. GARVAN where Pk,ℓ (x) = ⌊ℓ/2⌋ X ℓ(ℓ − m − 1)! xm k ℓ−2m . (ℓ − 2m)!m! m=0 Proof. Suppose k is odd and 1 ≤ ℓ ≤ k − 1 . First we prove that √ √ Pk,ℓ (x) = ( 12 k − 12 k 2 + 4x)ℓ + ( 12 k + 21 k 2 + 4x)ℓ . (4.11) (4.12) We have √ 1 √ 1 ⌊ℓ/2⌋   ℓ ( 12 k − 2 k 2 + 4x) + ( 12 k + 2 k 2 + 4x) = k ℓ−2j 21−ℓ (k 2 + 4x)j 2j j=0   ⌊ℓ/2⌋ ⌊ℓ/2⌋ j    ⌊ℓ/2⌋    X XX ℓ X j j  m ℓ−2m 2m−ℓ+1 ℓ  x k 2 . xm k ℓ−2m 22m−ℓ+1 = = m m 2j 2j m=0 m=0 j=0 j=m ℓ ℓ X The result (4.12) now follows from the binomial coefficient identity ⌊ℓ/2⌋    X ℓ ℓ(ℓ − m − 1)! j = 2ℓ−2m−1 , m 2j (ℓ − 2m)!m! j=m (4.13) which we leave as an exercise. We observe that if x = km + m2 + k 2 n(n + 1) + 2mnk then 1 k 2 1 k 2 − + √ k 2 + 4x = (k + 2m + 2kn)2 , 1 k2 2 √ 1 k2 2 + 4x = −kn − m, + 4x = k(n + 1) + m, and we see that
Dℓ ζ −kn−m + ζ k(n+1)+m = (−kn−m)ℓ +(k(n+1)+m)ℓ = Pk,ℓ (km+m2 +k 2 n(n+1)+2mnk). Similarly we find that
Dℓ ζ kn+m + ζ −kn+k−m = (kn+m)ℓ +(−kn+k−m)ℓ = Pk,ℓ (−km+m2 +k 2 n(n−1)+2mnk). We note that

Hk∗ q kn(n+1)/2+mn z m = (km + m2 + k 2 n(n + 1) + 2mnk) q kn(n+1)/2+mn z m ,

Hk∗ q kn(n−1)/2+mn z −m = (−km + m2 + k 2 n(n − 1) + 2mnk) q kn(n−1)/2+mn z −m . Thus and

Dℓ q kn(n+1)/2+mn z m (ζ −kn−m + ζ kn+k+m) = Pk,ℓ (Hk∗ ) q kn(n+1)/2+mn z m ,

Dℓ q kn(n+1)/2+mn z m (ζ kn+m + ζ −kn+k−m) = Pk,ℓ (Hk∗ ) q kn(n−1)/2+mn z −m . The result (4.10) follows from equations (4.8) and (4.9). Next we calculate D2m of each term in (4.1).  RANK-CRANK TYPE PDES 15 4.2. The term Ym (ζ, z, q). It is clear that the function Ym (ζ, z, q) has a zero of order 2m at ζ = 1. It is well-known that D2m (f (ζ)) = 2m X S(2m, j)f (j) (1), i=1 where the numbers S(2m, j) are Stirling numbers of the second kind. Since S(2m, 2m) = 1 it follows that D2m (Ym (ζ, z, q))) = Ym(2m) (1, z, q) = (−1)m−1 (2m)! (m+1)! (m−1)! [C ∗ (z, q)]2m+1 (q)2m−1 ∞ (4.14) by an easy calculation. 4.3. The term F0,m (ζ, q). By logarithmic differentiation we have δζ F0,m (ζ, q) = L0,m (ζ, q) F0,m(ζ, q). (4.15) where  ∞  X ζ m+1 q i ζ −m−1 q i L0,m (ζ, q) = K0,m (ζ) − (m + 1) − 1 − ζ m+1 q i 1 − ζ −m−1q i i=1  ∞  X ζ −m q i ζ mqi − +m 1 − ζ mq i 1 − ζ −m q i i=1 X X = K0,m (ζ) − (m + 1) (ζ m+1q i )n − (ζ −m−1 q i )n + m (ζ m q i )n − (ζ −mq i )n i,n≥1 i,n≥1 (4.16) K0,m (ζ) = m + Jm (ζ) − Jm−1 (ζ), m X nζ n Jm (ζ) = n=1 m X (4.17) . ζ (4.18) n n=0 For any positive integer k we define G2k ∞ X 1 B2k n2k−1 q n := G2k (q) := ζ(1 − 2k) + =− + Φ2k−1 (q), n 2 1−q 4k n=1 where B2n is the (2n)-th Bernoulli number, and ∞ X Φ2k−1 := Φ2k−1 (q) := σ2k−1 (n)q n . (4.19) (4.20) n=1 The function G2k is a normalized Eisenstein series. For k > 1 it is an entire modular form of weight 2k. We need the following Lemma 4.2. If m and a are positive integers, then
Ba+1 (m + 1)a+1 − 1 . Da (Jm (ζ)) = a+1 16 SONG HENG CHAN, ATUL DIXIT AND FRANK G. GARVAN Proof. Suppose m and a are positive integers. It is well-known that ∞ X x Bk xk = . ex − 1 k=0 k! (4.21) By taking the logarithmic derivative of (ζ m+1 − 1)/(ζ − 1) we find that 1 1 − . Jm (ζ) = m + (m + 1) m+1 ζ −1 ζ −1 Hence by (4.21) we have ∞ X
Bk+1 x (m + 1)k+1 − 1 xk . Jm (e ) = m + (k + 1)! k=0 The result now follows since Da (Jm (ζ)) =  d dx a Jm (ex ) . x=0 Corollary 4.3. Suppose a, m are integers a ≥ 0 and m ≥ 1.  a+1  − (m + 1)a+1 )Ga+1 (q) 2(m Da (L0,m (ζ, q)) = m + 12  0  Then if a is odd, if a = 0, otherwise. (4.22) Proof. The proof of (4.22) when a = 0 is straightforward. Suppose a is even and positive. Then by Lemma 4.2 Da (L0,m (ζ, q)) = Da (K0,m (ζ)) Ba+1 = (−ma+1 + (m + 1)a+1 ) a+1 = 0, since the Bernoulli numbers Bk are zero when k > 1 is odd. Now suppose a is odd. Then again by Lemma 4.2 Da (L0,m (ζ, q)) = Da (K0,m (ζ)) + 2(ma+1 − (m + 1)a+1 )Φa (q) Ba+1 = (−ma+1 + (m + 1)a+1 ) + 2(ma+1 − (m + 1)a+1 )Φa (q) a+1 = 2(ma+1 − (m + 1)a+1 )Ga+1 (q).  By applying Da−1 to both sides of (4.15) and using (4.22) we obtain the following recurrence Da (F0,m (ζ, q)) = (m + 12 )Da−1 (F0,m (ζ, q))  ⌊a/2⌋  X a−1 + (m2i − (m + 1)2i )G2i (q)Da−2i (F0,m (ζ, q)). 2 2i − 1 i=1 (4.23) RANK-CRANK TYPE PDES 17 This together with the initial value D0 (F0,m (ζ, q)) = F0,m (1, q) = m+1 m (4.24) uniquely determines the coefficients Da (F0,m (ζ, q)). We compute some examples 3 D0 (F0,2 ) = , 2 15 D1 (F0,2 ) = , 4 75 − 15G2 = 10 − 15Φ1 , D2 (F0,2 ) = 8 365 225 225 225 D3 (F0,2 ) = − G2 = − Φ1 , 16 2 8 2 1875 1125 − G2 + 450G22 − 195G4 = 82 − 600Φ1 + 450Φ21 − 195Φ3 . D4 (F0,2 ) = 32 2 4.4. The terms Fj,m (ζ, q) (1 ≤ j ≤ m − 1). Suppose 1 ≤ j ≤ m − 1. We may obtain a similar recurrence for Da (Fj,m (ζ, q)). This time we find that δζ Fj,m (ζ, q) = Lj,m (ζ, q) Fj,m(ζ, q), (4.25) for some function Lj,m (ζ, q) that satisfies  P j a+1  − (m − i)a+1 ) Ga+1 (q) if a is odd, 2 i=1 ((m
+ i + 1) Da (Lj,m (ζ, q)) = −j m + 21 (4.26) if a = 0,  0 otherwise. The proof of (4.26) is analogous to the proof of Corollary 4.3. By applying Da−1 to both sides of (4.25) and using (4.26) we obtain the following recurrence Da (Fj,m (ζ, q)) = −j(m + 12 )Da−1 (Fj,m (ζ, q)) ⌊a/2⌋ j X X a − 1 ((m + k + 1)2i − (m − k)2i )G2i (q)Da−2i (Fj,m (ζ, q)). 2 + 2i − 1 i=1 k=1 (4.27) This together with the initial value D0 (Fj,m (ζ, q)) = Fj,m(1, q) = (−1) j j Y i=1 m−i m+i+1 (4.28) 18 SONG HENG CHAN, ATUL DIXIT AND FRANK G. GARVAN uniquely determines the coefficients Da (Fj,m (ζ, q)). We compute some examples 1 D0 (F1,2 ) = − , 4 5 D1 (F1,2 ) = , 8 25 15 5 15 D2 (F1,2 ) = − − G2 = − − Φ1 , 16 2 4 2 25 225 125 225 + G2 = + Φ1 , D3 (F1,2 ) = 32 4 16 4 625 1125 255 1 255 D4 (F1,2 ) = − − G2 − 675G22 − G4 = − 225Φ1 − 675Φ21 − Φ3 . 64 4 2 4 2 4.5. The terms S2m+1 (ζ j+1, z, q) (1 ≤ j ≤ m−1). Again suppose that 1 ≤ j ≤ m−1. Consider the operator Tj that operates on a function f (ζ) by Tj (f (ζ)) = f (ζ j ). Then δζ ◦ Tj = j (Tj ◦ δζ ) . (4.29)
δζa ◦ Tj = j a Tj ◦ δζa , (4.30) Da ◦ Tj = j a Da . (4.31) ∗ Dℓ S2m+1 (ζ j+1, z, q) = (j + 1)ℓ P2m+1,ℓ (H2m+1 )Σ(2m+1) (z, q). (4.32) A simple induction argument gives and Thus by (4.10) we have 4.6. The main theorem. We are now ready to derive our main theorem. Applying D2m to both sides of (4.1) and using (4.10), (4.14), (4.23), (4.24), (4.27), (4.28), and (4.32) we have Theorem 4.4. (−1)m+1 (2m)! (m + 1)! (m − 1)![C ∗ (z, q)]2m+1 (q)2m+1 ∞ ∗ P2m+1,2m (H2m+1 ) = + m−1 2m XX (j + 1) 2m−a j=1 a=0   2m ∗ Da (Fj,m (ζ, q))P2m+1,2m−a (H2m+1 ) a ! − D2m (F0,m (ζ, q)) Σ(2m+1) (z, q) (4.33) where the coefficient functions Da (Fj,m )(ζ, q)) (0 ≤ j ≤ m − 1) are given recursively by (4.23) and (4.27), and their initial values (4.24) and (4.28). For n ≥ 0 let Vn be the Q-vector space spanned by the monomials Φa1 Φb3 Φc5 with a + 2b + 3c = n. We Define n X Vn ; (4.34) Wn = k=0 RANK-CRANK TYPE PDES 19 i.e., Wn is the Q-vector space spanned by the monomials Φa1 Φb3 Φc5 with 0 ≤ a+2b+3c ≤ n. We call Wn the space of quasi-modular forms of weight less than or equal to 2n. This agrees with the definition in [2, p.355] except this time we allow monomials of weight 0. Corollary 4.5. Suppose m ≥ 1. Then there exist quasi-modular forms fj ∈ Wj for 1 ≤ j ≤ m such that ! m−1 X ∗m ∗k Σ(2m+1) (z, q) = (2m)! [C ∗ (z, q)]2m+1 (q)2m+1 . (4.35) H2m+1 + fm−k H2m+1 ∞ k=0 Proof. Suppose m ≥ 1. It is well-known that Φ2n−1 ∈ Wn . See [2, Eq.(3.25)]. Equation (4.19), the recurrence (4.27) and a simple induction argument imply that Da (Fj,m (ζ, q)) ∈ W⌊a/2⌋ , for 1 ≤ j ≤ m − 1. Similarly using (4.22) and (4.23) we find that D2m (F0,m (ζ, q)) ∈ Wm . (4.36) ∗k H2m+1 Now we calculate the coefficient of in the right side of (4.33). The degree of the polynomial P2m+1,2m−a (x) is ⌊(2m − a)/2⌋. Assuming k ≤ ⌊(2m − a)/2⌋ we have 2k ≤ 2m − a and ⌊a/2⌋ ≤ m − k, and in this case Da (Fj,m (ζ, q)) is in Wm−k . This ∗k together with (4.36) implies that the coefficient of H2m+1 is in Wm−k for 0 ≤ k ≤ m. ∗m The coefficient of H2m+1 is f0 = 2 + 2 m−1 X j (−1) (j + 1) 2m j=1 (m − 1)!(m + 1)! (m − j − 1)!(m + j + 1)! j=0   m−1 2m 2(m − 1)!(m + 1)! X j 2m . (−1) (j + 1) = m − j − 1 (2m)! j=0 =2 We show that m−1 X j Y (m − k) m+k+1 k=1 (−1)j (j + 1)2m f0 = (−1)m+1 (m + 1)! (m − 1)!. In view of (4.37) this is equivalent to showing that   m−1 2m 2 X m+j+1 2m = 1, (−1) (j + 1) m−j −1 (2m)! j=0 (4.37) (4.38) (4.39) which we can rewrite as 2 m−1 X j=0 j (−1) (m − j) by replacing j by m − j − 1 in the sum. 2m   2m = (2m)!, j (4.40) 20 SONG HENG CHAN, ATUL DIXIT AND FRANK G. GARVAN Since m−1 X j=0 j (−1) (m − j) 2m     2m X 2m j 2m 2m , = (−1) (m − j) j j j=m+1 it suffices to prove 2m X j=0 j (−1) (m − j) 2m   2m = (2m)!. j (4.41) Beginning with the elementary identity  2m  X 2m j 2m−j xy = (x + y)2m , j j=0 setting x = − √1ζ and y = √ ζ, we obtain  2m  X 2m j=0 j (−1)j ζ m−j = ζ −m(ζ − 1)2m . (4.42) We apply D2m to both sides of (4.42) and argue as in Section 4.2 to obtain (4.41) which completes the proof of (4.38). The final result (4.35) follows from (4.33) by dividing both sides by f0 and using (4.38).  4.7. Some examples. We illustrate Theorem 4.4 and Corollary 4.5 with some examples. We show details of the calculations for the cases m = 1, 2. In cases m = 3, 4 we give the quasi-modular forms fj (1 ≤ j ≤ m) in Corollary 4.5, in terms of the functions Φ2k−1 (q) rather than the G2k (q). Example m = 1. ! 4 [C ∗ (z, q)]3 (q)3∞ = = = 9 + 2 H3 − D2 (F0,1 (ζ, q)) Σ(3) (z, q) ! 9 + 2 H3 − (5 − 12 Φ1 ) Σ(3) (z, q) ! 2 H3 + 4 + 12 Φ1 Σ(3) (z, q), and ! H3 + 2 + 6 Φ1 Σ(3) (z, q) = 2 [C ∗ (z, q)]3 (q)3∞ . This identity implies the Rank-Crank PDE (1.7) as in [2, Section 2]. RANK-CRANK TYPE PDES 21 Example m = 2. − 144 [C ∗(z, q)]5 (q)5∞ = 625 + 100 H∗5 + 2 H∗ 25 + 16 D0 (F1,2 (ζ, q)) (625 + 100 H∗5 + 2 H∗ 25 ) + 32 D1 (F1,2 (ζ, q)) (125 + 15 H∗5 ) + 24 D2(F1,2 (ζ, q)) (25 + 2 H∗5 ) + 40 D3 (F1,2 (ζ, q)) ! + 2 D4 (F1,2 (ζ, q)) − D4 (F0,2 (ζ, q)) Σ(5) (z, q) = 625 + 100 H∗5 + 2 H∗ 25 − 4 (625 + 100 H∗5 + 2 H∗ 25 ) + 20 (125 + 15 H∗5 ) − (30 + 180 Φ1 )(25 + 2 H∗ 5 ) + ( 125 + 2250 Φ1 ) 2 ! 1 + ( − 450 Φ1 − 1350 Φ21 − 255 Φ3) + (−82 + 600 Φ1 − 450Φ21 + 195 Φ3 ) Σ(5) (z, q), 2 and ! H∗ 25 + (60Φ1 + 10) H∗ 5 + 300 Φ21 + 10 Φ3 + 350 Φ1 + 24 Σ(5) (z, q) = 24 [C ∗ (z, q)]5 (q)5∞ . (4.43) In this case of Corollary 4.5 we see that f1 = 60Φ1 + 10, f2 = 300 Φ21 + 10 Φ3 + 350 Φ1 + 24. We show how this identity implies (1.18). We need the results δq ((q)∞ ) = −Φ1 (q)∞ , δq (Φ1 ) = 61 Φ1 − 2Φ21 + 65 Φ3 . This implies that H5∗ (Σ∗ (z, q)) = H5∗ ((q)3∞ G(5) (z, q)) = (q)3∞ (H5∗ − 30Φ1 )G(5) (z, q), (4.44) and H5∗ 2 (Σ∗ (z, q)) = H5∗ 2 ((q)3∞ G(5) (z, q)) = (q)3∞ (H5∗ 2 −60Φ1 H5∗ −50Φ1 +1500Φ21 −250Φ3 )G(5) (z, q). (4.45) Substituting (4.44), (4.45) into (4.43) we find (H5∗ 2 + 10H5∗ + 24 − 240Φ3 )G(5) (z, q) = 24 [C ∗(z, q)]5 (q)5∞ , which simplifies to (1.18) since H∗ = H5∗ + 5. 22 SONG HENG CHAN, ATUL DIXIT AND FRANK G. GARVAN Example m = 3. We find that f1 = 210 Φ1 + 28 f2 = 210 Φ3 + 8820 Φ21 + 252 + 4410 Φ1 f3 = 41160 Φ31 + 2450 Φ3 + 14 Φ5 + 720 + 22736 Φ1 + 2940 Φ3 Φ1 + 102900 Φ21 Example m = 4. We find that f1 = 504 Φ1 + 60 f2 = 1260 Φ3 + 24948 Φ1 + 1308 + 68040 Φ21 f3 = 136080 Φ3 Φ1 + 504 Φ5 + 45360 Φ3 + 2449440 Φ21 + 403704 Φ1 + 2449440 Φ31 + 12176 f4 = 40320 + 2126232 Φ1 + 404082 Φ3 + 9828 Φ5 + 2653560 Φ3 Φ1 + 21820428 Φ21 + 9072 Φ1 Φ5 + 47764080 Φ31 + 18 Φ7 + 1224720 Φ3 Φ21 + 11340 Φ23 + 11022480 Φ41. 5. Concluding remarks The main goal of this paper was to show how the generalized Lambert series identity (1.3) leads to the higher level Rank-Crank-type PDEs of Zwegers. The first author’s proof [8] of (1.3) only involves a partial fraction argument and this together with the proof in Section 4 gives an elementary q-series proof of these higher level Rank-Cranktype PDEs. The elliptic function proof of (1.3) in Section 2 is independent of the other sections. Our form of Zwegers’s result (1.23) was given above in (4.35). In our form the coefficients are quasimodular forms rather than holomorphic modular forms. The quasimodular function E2 occurs in Zwegers’s result as part of the definition of his operator Hk . Our coefficient functions are given recursively. It would be interesting to find explicit expressions for the coefficients and to derive the form of Zwegers’s result by our method. The coefficients in the 4th order PDE (1.18) only involve the holomorphic modular form E4 , and the differential operator H∗ does not involve the quasimodular E2 . It would be interesting to determine whether there is a renormalization of higher order Rank-Crank-type PDEs which only involve holomorphic modular forms, either as coefficients or in the definition of the differential operator. Bringmann, Lovejoy and Osburn [5], [6] found Rank-Crank-type PDEs for overpartitions. Bringmann and Zwegers [7] showed how these results fit into the framework of non-holomorphic Jacobi forms and found an infinite family of these PDEs. However these PDEs only involve Appell functions of level 1 or 3. It would be interesting to determine whether the methods of this paper could be extended to find PDEs for higher level analogues. Acknowledgements We would like to thank Bruce Berndt and Ken Ono for their comments and suggestions. References [1] G.E. Andrews and F.G. Garvan, Dyson’s crank of a partition, Bull. Amer. Math. Soc. (N.S.) 18 (1988), 167–171. RANK-CRANK TYPE PDES 23 [2] A.O.L. Atkin and F.G. Garvan, Relations between the ranks and cranks of partitions, Ramanujan J. 7 (2003), 343–366. [3] A.O.L. Atkin and H.P.F. Swinnerton-Dyer, Some properties of partitions, Proc. London Math. Soc. (3) 4 (1954), 84–106. [4] B.C. Berndt, Number Theory in the Spirit of Ramanujan, American Mathematical Society, Providence, RI, 2006. [5] K. Bringmann, J. Lovejoy and R. Osburn, Rank and crank moments for overpartitions, J. Number Theory 129 (2009), 1758-1772. [6] K. Bringmann, J. Lovejoy and R. Osburn, Automorphic properties of generating functions for generalized rank moments and Durfee symbols, Int. Math. Res. Not. (2010), no. 2, 238–260. [7] K. Bringmann and S. Zwegers, Rank-crank type PDE’s and non-holomorphic Jacobi forms, Math. Res. Lett. 17 (2010), 589–600. [8] S.H. Chan, Generalized Lambert series identities, Proc. London Math. Soc. (3) 91 (2005), 598– 622. [9] F.J. Dyson, Some guesses in the theory of partitions, Eureka (Cambridge) 8 (1944), 10–15. [10] F.J. Dyson, Selected papers of Freeman Dyson with commentary, Amer. Math. Soc., Providence, RI, 1996. [11] A. Erdélyi, W. Magnus, F. Oberhettinger and F.G. Tricomi, Higher Transcendental Functions. Vol. II, McGraw-Hill Book Company, New York, 1953. [12] F.G. Garvan, New Combinatorial Interpretations of Ramanujan’s Partition Congruences Mod 5,7 and 11, Trans. Amer. Math. Soc. 305 (1988), 47–77. [13] F.G. Garvan, Generalizations of Dyson’s ranks and non-Rogers-Ramanujan partitions, Manuscr. Math. 84 (1994), 343–359. [14] M. Jackson, On some formulae in partition theory and bilateral basic hypergeometric series, J. London Math. Soc. 24 (1949), 233–237. [15] R.P. Lewis, Relations between ranks and cranks modulo 9, J. London Math. Soc. (2) 45 (1992), 222–231. [16] S. Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa, New Delhi, 1988. [17] J. Tannery and J. Molk, Éléments de la Théorie des Fonctions Elliptiques, Vol. III, GauthierVillars, Paris, 1896 [reprinted: Chelsea, New York, 1972] [18] G.N. Watson, The final problem: An account of the mock theta functions, J. London Math. Soc. 11 (1936), 55-80. [19] E.T. Whittaker and G.N. Watson, A Course of Modern Analysis, 4th ed., Cambridge University Press, Cambridge, 1966. [20] S.P. Zwegers, Mock Theta Functions, Ph.D. Thesis, Universiteit Utrecht, 2002. [21] S.P. Zwegers, Rank-Crank type PDE’s for higher level Appell functions, Acta Arith. 144 (2010), 263–273. Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang link, Singapore, 637371, Republic of Singapore E-mail address: ChanSH@ntu.edu.sg Department of Mathematics, University of Illinois, 1409 West Green Street, Urbana, IL 61801, USA E-mail address: aadixit2@illinois.edu Department of Mathematics, University of Florida, Gainesville, Florida 32611, USA E-mail address: fgarvan@ufl.edu