RELATIONS BETWEEN THE RANKS

AND CRANKS OF PARTITIONS

A.O.L. Atkin and F.G. Garvan

Dedicated to the memory of Robert A. Rankin

Abstract. New identities and congruences involving the ranks and cranks of partitions are proved. The proof depends on a new partial differential equation connecting their generating functions.

1. Introduction

Dyson [D1], [D3, p.52] defined the rank of a partition as the largest part minus the number of parts. Let N(m, n) denote the number of partitions of n with rank m, then

(1.1)  ∑_{m} N(m, n) = p(n),

the number of partitions of n; and

(1.2)  N(m, n) = N(−m, n),

using the classical conjugacy of partitions.

Andrews and Garvan [A-G] defined the crank of a partition. It is the largest part if the partition contains no ones, and is otherwise the number of parts larger than the number of ones minus the number of ones. Let M(m, n) denote the number of partitions of n with crank m, then

(1.3)  ∑_{m} M(m, n) = p(n),

the number of partitions of n; and

(1.4)  M(m, n) = M(−m, n),

for n > 1. A direct combinatorial proof of (1.4) was found recently by Berkovich and Garvan [B-G].

Typeset by AMS-TEX

1991 Mathematics Subject Classification. Primary 11P81, 11P83; Secondary 05A17, 05A19, 11F11, 11F25, 11F33.

Key words and phrases. partitions, rank, crank, Ramanujan congruences, Eisenstein series, modular forms, derivatives.

Research of the second author supported in part by the NSF under grant number DMS-9870052.

We now state the generating function R(z, q) for the rank. We have

(1.5)  R(z, q) = ∑_{n≥0} ∑_{m} N(m, n)z^{m}q^{n}

= 1 + ∑_{n=1}^{∞} \frac{q^{n^{2}}}{(zq)_{n}(z^{−1}q)_{n}} , (1.6)

and

(1.7)  ∑_{n≥0} N(m, n)q^{n} = \frac{1}{(q)_{∞}} ∑_{n=1}^{∞}(−1)^{n−1}q^{\frac{n}{2} (3n−1)+|m|n}(1 − q^{n}).

Here we are using the notation

(1.8)  (a)_{n} = (a; q)_{n} = (1 − a)(1 − aq) ⋯ (1 − aq^{n−1}),
(q)_{∞} = lim_{n→∞}(q)_{n},

where |q| < 1.

Below we state the generating function C(z, q) for the crank. If we amend the definition of M(m, n) for n = 1, then the generating function can be given as an infinite product. Accordingly, throughout this paper we assume

(1.9)  M(0, 1) = −1, M(−1, 1) = M(1, 1) = 1, and M(m, 1) = 0 otherwise.

Then we have

(1.10)  C(z, q) = ∑_{n≥0} ∑_{m} M(m, n)z^{m}q^{n}

= ∏_{n=1}^{∞} \frac{(1 − q^{n})}{(1 − zq^{n})(1 − z^{−1}q^{n})}, (1.11)

and

(1.12)  ∑_{n≥0} M(m, n)q^{n} = \frac{1}{(q)_{∞}} ∑_{n=1}^{∞}(−1)^{n−1}q^{\frac{n}{2} (n−1)+|m|n}(1 − q^{n}).

Equation (1.11) follows from [A-G, eq. (1.11)] and [A-G, Thm 1]. Equation (1.12) then follows from [G1, eq. (7.20)].

The main result of this paper is a fundamental partial differential equation (PDE) connecting the rank and crank generating functions. See Theorem 1.1 below. Let

(1.13)  R^{∗}(z, q) := \frac{R(z, q)}{(1 − z)},
(1.14)  C^{∗}(z, q) := \frac{C(z, q)}{(1 − z)}.

Define the differential operators

(1.15)  δ_{z} = z \frac{∂}{∂z} ,  δ_{q} = q \frac{∂}{∂q} .

Theorem 1.1. We have

(1.16)  z(q)_{∞}^{2} [C^{∗}(z, q)]^{3} = (3δ_{q} + \frac{1}{2}δ_{z} + \frac{1}{2}δ_{z}^{2}) R^{∗}(z, q),

and

(1.17) z(q)_{∞}^{2} [C(z, q)]^{3} = (3(1 − z)^{2}δ_{q} + \frac{1}{2}(1 − z)^{2}δ_{z}^{2} − \frac{1}{2}(z^{2} − 1)δ_{z} + z) R(z, q).

We prove equation (1.16) in Section 2. The Rank-Crank PDE (1.17) follows easily from (1.16) by using (1.13) and (1.14).

Let N(k, t, n) denote the number of partitions of n with rank congruent to k modulo t. Then for t = 5 or t = 7

(1.18)  N(k, t, n) = \frac{1}{t} p(n),  0 ≤ k ≤ t − 1;

for all n satisfying 24n ≡ 1 (mod t). These combinatorial results immediately imply Ramanujan’s partition congruences

(1.19)  p(5n + 4) ≡ 0 (mod 5),
(1.20)  p(7n + 5) ≡ 0 (mod 7).

Let M(k, t, n) denote the number of partitions of n with crank congruent to k modulo t. Then for t = 5, t = 7, or t = 11

(1.21)  M(k, t, n) = \frac{1}{t} p(n),  0 ≤ k ≤ t − 1;

for all n satisfying 24n ≡ 1 (mod t). These combinatorial results again imply Ramanujan’s partition congruences mod 5 and mod 7, and in addition his congruence

(1.22)  p(11n + 6) ≡ 0 (mod 11).

There are many more rank identities. For example,

(1.23)  N(1, 5, 5n + 1) = N(2, 5, 5n + 1),

and others for the moduli 5, 7, 8, 9, and 12. The results for 5 and 7 were all found by Dyson [D1], [D3, p.53] and proved by Atkin and Swinnerton-Dyer [A-SD]. The results for 8, 9, and 12, were found by Lewis [L1] and subsequently proved by Lewis and Santa-Gadea in a series of papers [SG1], [SG2], and [L-SG].

There are similar identities for the crank. For example,

(1.24) M(0, 8, 2n + 1) + M(1, 8, 2n + 1) = M(3, 8, 2n + 1) + M(4, 8, 2n + 1),

and others for the moduli 5, 7, 8, 9, 10, and 11. These were proved in [G1], [G2], and [G3].

There are identities between the rank and the crank. For example,

(1.25)  M(4, 9, 3n) = N(4, 9, 3n),

and others for the moduli 5, 7, 8, and 9. These results were proved in [G2], [G3], [L1], [L3], [L-SG1], and [SG1].

In this paper we consider linear relations modulo a prime p. There are congruences for the rank to the moduli 11 and 13. For example, (1.26) 2N(2, 11, 11n) + N(3, 11, 11n) + 7N(4, 11, 11n) + N(5, 11, 11n) ≡ 0 (mod 11).

Results of this type are due to Atkin and Hussain [A-H] and O’Brien [OB].

It is a surprising fact that there is one analogous relation for the crank for every prime p. This was a mystery to us until we realized that these crank congruences follow from the identity

(1.27)  ∑_{k=1}^{n} k^{2} M(k, n) = n p(n),

due to Dyson [D2], who gave a combinatorial proof.

There is an extra linear congruence for the crank modulo p for p = 41, 53, 83, and 120667369.

For each prime p > 13 there are seven congruences involving both the rank and the crank modulo p. For example,

(1.28) 6N(0, 29, 29n + 23) + 17N(1, 29, 29n + 23) + 24N(2, 29, 29n + 23)

+ 18N(3, 29, 29n + 23) + 17N(4, 29, 29n + 23) + 14N(5, 29, 29n + 23)

+ 22N(6, 29, 29n + 23) + 24N(7, 29, 29n + 23) + 2N(9, 29, 29n + 23)

+ 15N(10, 29, 29n + 23) + 19N(11, 29, 29n + 23) + 18N(12, 29, 29n + 23)

+ 20N(13, 29, 29n + 23) + 16N(14, 29, 29n + 23)

≡ 11M(0, 29, 29n + 23) + 17M(1, 29, 29n + 23) + 28M(2, 29, 29n + 23) + 26M(4, 29, 29n + 23) + 6M(5, 29, 29n + 23) + 28M(8, 29, 29n + 23)

(mod 29).

These congruences come from certain exact relations between the rank and the crank.

For even j ≥ 2, we define

(1.29)  N_{j}(n) = ∑_{k} k^{j} N(k, n),
(1.30)  M_{j}(n) = ∑_{k} k^{j} M(k, n).

The following is the simplest exact relation

(1.31)  N_{4}(n) = −(2n + \frac{2}{3}) M_{2}(n) + \frac{8}{3}M_{4}(n) + (1 − 12n) N_{2}(n).

In fact, there are polynomials P_{k}(n) of degree k − 1 and Q_{k,j}(n) of degree k − j (for 1 ≤ j ≤ k) such that

(1.32)  N_{2k}(n) = P_{k}(n) N_{2}(n) + ∑_{j=1}^{k} Q_{k,j}(n) M_{2j}(n),

for k = 2, 3, 4, and 5. For k = 6 there is no such relation. For k = 7 there is a similar relation but with an extra term N_{12}(n). See Theorem 5.1 below. The proof of these exact relations depends on the Rank-Crank PDE (1.17).

We call the functions N_{j} and M_{j} (defined in (1.29), (1.30)) rank and crank moments, respectively. We define the following generating functions

(1.33)  R_{j} = R_{j}(q) = ∑_{n≥1} N_{j}(n)q^{n},
(1.34)  C_{j} = C_{j}(q) = ∑_{n≥1} M_{j}(n)q^{n},

for even j. We find that

(1.35)  δ_{z}^{j}R(z, q)|_{z=1} = \{ \begin{matrix}R_{j}, & j even, \\ 0, & j odd, \end{matrix}

using (1.2) and (1.5). Similarly we find that

(1.36)  δ_{z}^{j}C(z, q)|_{z=1} = \{ \begin{matrix}C_{j}, & j even, \\ 0, & j odd. \end{matrix}

In Section 2 we show how the Rank-Crank PDE (1.17) follows from a certain elliptic-function identity (2.3). In Section 3 we prove some results for Eisenstein series, modular forms and quasi-modular forms. In Section 4 we show how the crank moment functions can be written in terms of Eisenstein series, and we derive some results for the derivatives of crank moment functions. In Section 5 we show how the Rank-Crank PDE and certain results for the derivatives of Eisenstein series lead to exact relations between rank and crank moments. As a bonus we show that the 23rd power of the Dedekind eta-function can be written in terms of rank and crank moments. In Section 6 we consider congruence relations between rank and crank moments.

2. Proof of the rank-crank PDE

The rank-crank PDE follows easily from an identity in [A-SD]. Define

(2.1)  J(z, q) := ∏_{n=1}^{∞}(1 − z^{−1}q^{n})(1 − zq^{n−1}),

and

(2.2)  S(z, ζ, q) := ∑_{n=−∞}^{∞} \frac{(−1)^{n}ζ^{n}q^{3n(n+1)/2}}{1 − zq^{n}} .

Then (2.3)

ζ^{3}S(zζ, ζ^{3}, q) + S(zζ^{−1}, ζ^{−3}, q) − ζ \frac{J(ζ^{2}, q)}{J(ζ, q)}S(z, 1, q) = \frac{J(ζ, q)J(ζ^{2}, q)(q)_{∞}^{2}}{J(ζz, q)J(z, q)J(zζ^{−1}, q)}.

This identity is equation (5.1) in [A-SD, p.94] and was one of the key identities required to prove Dyson’s results for the rank modulo 5 and 7. We let g(ζ) denote either side of (2.3) as a function of ζ. By considering the right side of (2.3) we see that g(ζ) has a double zero at ζ = 1 and that

(2.4)  g′′(1) = 4(q)_{∞}^{3} [C^{∗}(z, q)]^{3} ,

where C^{∗}(z, q) is defined in (1.14). We let h(ζ) be the sum of the first two terms on the left side of (2.3); i.e.,

(2.5)  h(ζ) = ζ^{3}S(zζ, ζ^{3}, q) + S(zζ^{−1}, ζ^{−3}, q).

We find that

h′′(1) = ∑_{n=−∞}^{∞}(−1)^{n}q^{3n(n+1)/2} (\frac{6(3n^{2} + 3n + 1)}{1 − zq^{n}} + \frac{4z(3n + 2)q^{n}}{(1 − zq^{n})^{2}} + \frac{4z^{2}q^{2n}}{(1 − zq^{n})^{3}} )

(2.6) = (6 + 12δ_{q} + 6δ_{z} + 2δ_{z}^{2})S(z, 1, q).

We let j(ζ) be the third term on the right side of (2.3)

(2.7)  j(ζ) = −ζ \frac{J(ζ^{2}, q)}{J(ζ, q)}S(z, 1, q).

We find that

(2.8) j′′(1) = −2 (1 − 6 ∑_{n≥1} \frac{q^{n}}{(1 − q^{n})^{2}} ) S(z, 1, q) = −2(1 − 6Φ_{1}(q))S(z, 1, q),

where

(2.9)  Φ_{1}(q) = ∑_{n=1}^{∞} \frac{nq^{n}}{1 − q^{n}} .

RELATIONS BETWEEN THE RANK AND THE CRANK 7

The functions Φ_{j} are defined below in (3.1). We define

(2.10)  P (q) = ∑_{n=0}^{∞} p(n)q^{n} = \frac{1}{(q)_{∞}} .

By differentiating logarithmically with respect to q we obtain the well-known identity

(2.11)  δ_{q}P (q) = Φ_{1}(q) P (q),

or

(2.12)  δ_{q}(q)_{∞} = −Φ_{1}(q)(q)_{∞}.

The following identity is equation (7.10) in [G1]:

(2.13)  zS(z, 1, q) = (q)_{∞}(−1 + R^{∗}(z, q)).

We apply δ_{q} to both sides of (2.13) and use (2.12) to find that

(2.14)  zδ_{q}S(z, 1, q) = (q)_{∞}δ_{q}R^{∗}(z, q) − zΦ_{1}(q)S(z, 1, q).

Similarly we find that

(2.15)  zδ_{z}S(z, 1, q) = (q)_{∞}δ_{z}R^{∗}(z, q) − zS(z, 1, q),
(2.16)  zδ_{z}^{2}S(z, 1, q) = (q)_{∞} (δ_{z}^{2} − 2δ_{z}) R^{∗}(z, q) + zS(z, 1, q).

Now

(2.17)  g′′(1) = h′′(1) + j′′(1).

Using (2.4), (2.6), (2.8), and (2.13)–(2.16) this equation becomes

(2.18)  z(q)_{∞}^{2} [C^{∗}(z, q)]^{3} = (3δ_{q} + \frac{1}{2}δ_{z} + \frac{1}{2}δ_{z}^{2}) R^{∗}(z, q).

From (1.13) and (1.14) we find that

(2.19)  δ_{z}R^{∗}(z, q) = \frac{δ_{z}R(z, q) + zR^{∗}(z, q)}{1 − z},
(2.20)  δ_{z}^{2}R^{∗}(z, q) = \frac{δ_{z}^{2}R(z, q) + 2zδ_{z}R^{∗}(z, q) + zR^{∗}(z, q)}{1 − z},
(2.21)  δ_{q}R^{∗}(z, q) = \frac{δ_{q}R(z, q)}{1 − z}.

Using (2.19)–(2.21) we can write (2.18) in terms of C(z, q) and R(z, q):

(2.22) z(q)_{∞}^{2} [C(z, q)]^{3} = (3(1 − z)^{2}δ_{q} + \frac{1}{2}(1 − z)^{2}δ_{z}^{2} − \frac{1}{2}(z^{2} − 1)δ_{z} + z) R(z, q),

which is the rank-crank PDE.

3. Eisenstein series, modular forms and derivatives

Following Ramanujan [Ram, p.163] we define

(3.1)  Φ_{j}(q) = ∑_{n=1}^{∞} \frac{n^{j}q^{n}}{1 − q^{n}} = ∑_{m,n≥1} n^{j}q^{nm} = ∑_{n=1}^{∞} σ_{j}(n)q^{n},

for j ≥ 1 odd and where σ_{j}(n) = ∑_{d|n} d^{j}. As usual we let q = exp(2πiτ) where τ is in the complex upper half-plane H so that |q| < 1. For n even the Eisenstein series E_{n}(τ) is defined by

E_{n}(τ) = 1 − \frac{2n}{B_{n}} ∑_{k=1}^{∞} σ_{n−1}(k)e^{2πikτ} ,

(3.2)

= 1 − \frac{2n}{B_{n}} Φ_{n−1}(q),

where the Bernoulli numbers B_{n} are defined by

\frac{x}{e^{x} − 1} = ∑_{n=0}^{∞} B_{n} \frac{x^{n}}{n!} .

Ramanujan [R, p.140] considered in particular the Eisenstein series

E_{2} = 1 − 24 Φ_{1},
(3.3)  E_{4} = 1 + 240 Φ_{3},
E_{6} = 1 − 540 Φ_{5}.

For even n ≥ 4, E_{n} is a modular form of weight n for the full modular group Γ = SL_{2}(ℤ) [Ran]. E_{2} is not a modular form, but is transformed by the generators of Γ according to

(3.4)  E_{2}(τ + 1) = E_{2}(τ),
τ^{−2}E_{2}(−1/τ) = E_{2}(τ) + \frac{12}{2πiτ} .

See [K, p.113]. Ramanujan [Ram, p.165] found that

δ_{q}(E_{2}) = \frac{E_{2}^{2} − E_{4}}{12} ,
(3.5)  δ_{q}(E_{4}) = \frac{E_{2}E_{4} − E_{6}}{3} ,
δ_{q}(E_{6}) = \frac{E_{2}E_{6} − E_{4}^{2}}{2} ,

where

δ_{q} = q \frac{d}{dq} .

More generally, it is known that if f is a modular form of weight k then

12δ_{q}(f) − kE_{2}f

is a modular form of weight (k + 2). See [SD, p.19].

Following Serre [S, p.88], we let M_{k} denote the vector space of modular forms of weight 2k. Then

(3.6)  dimM_{k} = \{ \begin{matrix}[k/6], & if k ≡ 1 (mod 6), \\ [k/6] + 1, & otherwise, \end{matrix}

and the set

(3.7)  {E_{4}^{a}E_{6}^{b} : 2a + 3b = k with a and b nonnegative integers}

forms a basis for M_{k}. See Serre [S, p.88]. Thus any Eisenstein series E_{2n} (for n ≥ 2) can be written in terms of E_{4} and E_{6}. This can be done explicitly using well-known recurrences. See [Ap, pp.12-13] and [B, pp.331-332].

The graded algebra of modular forms is given by

(3.8)  M = ∑_{k=0}^{∞} M_{k} = ℂ[E_{4}, E_{6}].

See Serre [S,p.89]. We need to extend this algebra to include E_{2}. We say that f is a quasi-modular form if it is in the algebra generated by E_{2} and M. We extend the definition of weight by defining the weight of E_{2} to be 2. Let n be a nonnegative integer. Then the space of quasi-modular forms of weight ≤ 2n is

(3.9)  E_{n} = {∑_{j=0}^{n} f_{j}E_{2}^{j} : f_{j} ∈ ∑_{k=0}^{n−j} M_{k}} ,

which is clearly a vector space over ℂ. Below in Corollary 3.6 we give a basis for E_{n}.

Quasi-modular forms were first studied systematically by Kaneko and Zagier [K-Z]. We need some independence results for modular forms and quasi-modular forms. The results for quasi-modular forms follow from [K-Z; Proposition 1(b), p.167]. We have included proofs for completeness since they are elementary and the details of the relevant proof in [K-Z] are omitted.

Lemma 3.1. Let n be a nonnegative integer. Suppose that

(3.10)  f_{k} : H ⟶ ℂ, where f_{k}(τ + 1) = f_{k}(τ),  (0 ≤ k ≤ n)

for all τ ∈ H, and

(3.11)  ∑_{k=0}^{n} τ^{k}f_{k}(τ) = 0,

10 A.O.L. ATKIN AND F.G. GARVAN

for all τ ∈ H. Then

(3.12)  f_{0}(τ) = f_{1}(τ) = ⋯ = f_{n}(τ) = 0,

for all τ ∈ H. Proof. Suppose (3.10) and (3.11) hold. Let τ ∈ H be arbitrary but fixed. Then

(3.13)  ∑_{k=0}^{n}(τ + m)^{k}f_{k}(τ) = 0,

for all integers m. Hence the polynomial

p(z) = ∑_{k=0}^{n} f_{k}(τ)z^{k}

has infinitely many zeros. The result follows. □ Corollary 3.2. Non-zero modular forms of different weights are linearly independent over ℂ. Proof. Suppose that there are complex constants c_{k} such that

(3.14)  ∑_{k=0}^{n} c_{k}f_{k}(τ) = 0,

for all τ ∈ H, where f_{k}(τ) is a modular form of weight k. We apply τ → −1/τ to obtain

∑_{k=0}^{n} c_{k}τ^{k}f_{k}(τ) = 0,

for all τ ∈ H. The functions f_{k} satisfy (3.10) so by Lemma 3.1 we have

c_{0}f_{0}(τ) = c_{1}f_{1}(τ) = ⋯ = c_{n}f_{n}(τ) = 0,

and the result follows. □ Proposition 3.3. Let n be a nonnegative integer. Suppose that

(3.15)  f(τ) := ∑_{j=0}^{n} f_{j}(τ)E_{2}^{j}(τ) = 0,

for all τ ∈ H, and f is a quasi-modular form of weight ≤ 2n, so that each f_{j} is a sum of modular forms of weight ≤ 2n − 2j. Then

f_{0} = f_{1} = ⋯ = f_{n} = 0.

Proof. Suppose (3.15) holds and suppose for 0 ≤ j ≤ n

f_{j} = f_{j,0} + f_{j,4} + ⋯ + f_{j,2n−2j},

where each f_{j,k} is a modular form of weight k. Hence

(3.16)  ∑_{j=0}^{n} (f_{j,0} + ∑_{k=2}^{n−j} f_{j,2k}) E_{2}^{j} = 0.

Applying τ → −1/τ we obtain

(3.17)  ∑_{j=0}^{n} (f_{j,0} + ∑_{k=2}^{n−j} τ^{2k}f_{j,2k}) (τ^{2}E_{2} + ατ)^{j} = 0,

where

(3.18)  α = −\frac{6i}{π},

by (3.4). We rewrite (3.17) as

∑_{j=0}^{n} (f_{j,0} + ∑_{k=2}^{n−j} τ^{2k}f_{j,2k}) τ^{j} ∑_{ℓ=0}^{j} \binom{j}{ℓ}τ^{ℓ}E_{2}^{ℓ}α^{j−ℓ} = 0,

whence

(3.19) f_{0,0} + αf_{1,0}τ

+ ∑_{m=2}^{2n} ( ∑_{\begin{matrix}0≤j≤n \\ 0≤ℓ≤j \\ j+ℓ=m \end{matrix}} f_{j,0}\binom{j}{ℓ}E_{2}^{ℓ}α^{j−ℓ} + ∑_{\begin{matrix}0≤j≤n \\ 2≤k≤n−j \\ 0≤ℓ≤j \\ j+ℓ+2k=m \end{matrix}} \binom{j}{ℓ}α^{j−ℓ}f_{j,2k}E_{2}^{ℓ}) τ^{m} = 0.

Since the f_{j,k} and E_{2} satisfy (3.10), by Lemma 3.1 we have

f_{0,0} = 0,

f_{1,0} = 0, (3.20)

∑_{\begin{matrix}0≤j≤n \\ 0≤ℓ≤j \\ j+ℓ=m \end{matrix}} f_{j,0}\binom{j}{ℓ}E_{2}^{ℓ}α^{j−ℓ} + ∑_{\begin{matrix}0≤j≤n \\ 2≤k≤n−j \\ 0≤ℓ≤j \\ j+ℓ+2k=m \end{matrix}} \binom{j}{ℓ}α^{j−ℓ}f_{j,2k}E_{2}^{ℓ} = 0, 2 ≤ m ≤ 2n.

There are 2n+1 equations in (3.20). Each f_{j,2k} occurs in some equation. We prove that

(3.21)  f_{j,2k} = 0,

for all j, k such that f_{j,2k} occurs in the m-th equation of (3.20). We proceed by induction on m. The result holds for m = 0, and m = 1. Suppose it holds for m < M where M is a fixed positive integer ≤ 2n. We consider the M-th equation. It is clear that if a term f_{j,2k}E_{2}^{ℓ} with ℓ ≥ 1 occurs in this equation, then f_{j,2k} must have appeared in a previous equation, and so is zero by the induction hypothesis. The remaining terms in the M-th equation all have ℓ = 0, and are true modular forms, and it is easy to see that they have different weights. Hence they are all zero by Corollary 3.2, and thus the result holds for m = M, and so for all m with 0 ≤ m ≤ 2n by induction. □

Corollary 3.4. Let n be a nonnegative integer. The set

(3.22)  {E_{2}^{a}E_{4}^{b}E_{6}^{c} : a + 2b + 3c ≤ n with a, b, c nonnegative integers}

is a basis for the space of quasi-modular forms of weight ≤ 2n.

In the next section we shall see that the crank moment functions C_{a} can be written in terms of the Φ_{j} but are not modular forms. Thus we need some results for the Φ_{j}. Using (3.3) and (3.5) we find that

δ_{q}(Φ_{1}) = \frac{5}{6} Φ_{3} + \frac{1}{6} Φ_{1} − 2 Φ_{1}^{2},
(3.23)  δ_{q}(Φ_{3}) = \frac{7}{10} Φ_{5} + \frac{1}{3} Φ_{3} − \frac{1}{30} Φ_{1} − 8 Φ_{1}Φ_{3},
δ_{q}(Φ_{5}) = \frac{1}{2} Φ_{5} + \frac{1}{42} Φ_{1} − 12 Φ_{1}Φ_{5} + \frac{10}{21} Φ_{3} + \frac{400}{7} Φ_{3}^{2}.

We define the weight ω of a monomial Φ_{1}^{a}Φ_{3}^{b}Φ_{5}^{c} as

(3.24)  ω(Φ_{1}^{a}Φ_{3}^{b}Φ_{5}^{c}) = 2a + 4b + 6c.

It is clear from (3.3) that the monomial is a sum of quasi-modular forms of different weights, the form E_{2}^{a}E_{4}^{b}E_{6}^{c} having the highest weight ω. From (3.2), (3.3) and the fact the set in (3.7) forms a basis for M_{n}, we see that for n > 1 there are constants α_{b,c} such that

(3.25)  Φ_{2n−1} = ∑_{0<2b+3c≤n} α_{b,c}Φ_{3}^{b}Φ_{5}^{c}.

In the sum 2b + 3c is positive since Φ_{2n−1}(q) = 0 for q = 0. For example,

(3.26)  Φ_{11} = \frac{1}{13} (63(Φ_{3} + 240 Φ_{3}^{2} + 19200 Φ_{3}^{3} + 200 Φ_{5}^{2}) − 50Φ_{5}) .

For n ≥ 1 we let V_{n} be the ℂ-vector space spanned by monomials Φ_{1}^{a}Φ_{3}^{b}Φ_{5}^{c} with a + 2b + 3c = n. We define

(3.27)  W_{n} = ∑_{k=1}^{n} V_{n};

i.e., W_{n} is the ℂ-vector space spanned by monomials Φ_{1}^{a}Φ_{3}^{b}Φ_{5}^{c} with 0 < a+2b+3c ≤ n. Using (3.23) and induction we can show that

(3.28)  δ_{q} (W_{n}) ⊂ W_{n+1},

so that

(3.29)  δ_{q}^{m} (W_{n}) ⊂ W_{n+m},

for m ≥ 0.

Theorem 3.5. Let n be a nonnegative integer. The set

(3.30)  {Φ_{1}^{a}Φ_{3}^{b}Φ_{5}^{c} : 0 < a + 2b + 3c ≤ n with a, b, c nonnegative integers}

is a basis for W_{n}.

Proof. The set in (3.30) spans W_{n} by definition. We show that the set is linearly independent, by showing a slightly larger set is linearly independent. By equation (3.3) and Corollary 3.4 the set of monomials Φ_{1}^{a}Φ_{3}^{b}Φ_{5}^{c}, where 0 ≤ a + 2b + 3c ≤ n, spans E_{n} and the number of such monomials is equal to dim E_{n}. Hence, these monomials form a basis for E_{n} and are linearly independent. □

Corollary 3.6. For n ≥ 1

dim V_{n} = ∑_{k=0}^{n} dim M_{k}, (3.31)

dim W_{n} = n + ∑_{k=2}^{n}(n − k + 1)dim M_{k}. (3.32)

Proof. The result follows from Theorem 3.5 and the fact that the set in (3.7) forms a basis for M_{k}. □

We have the following table.

k dim M_{k} dim V_{k} dim W_{k}
1  0  1  1
2  1  2  3
3  1  3  6
4  1  4  10
5  1  5  15
6  2  7  22
7  1  8  30
8  2  10  40
9  2  12  52
10  2  14  66

4. Crank moments

We now prove that the crank moment functions C_{a} for a even can be written in terms of P and the Φ_{j}. From (1.11) we have

(4.1)  δ_{z}C(z, q) = L(z, q)C(z, q),

where

(4.2)  L(z, q) = ∑_{n≥1} (\frac{zq^{n}}{1 − zq^{n}} − \frac{z^{−1}q^{n}}{1 − z^{−1}q^{n}} ) ,

= ∑_{m,n≥1} (z^{m}q^{mn} − z^{−m}q^{mn}) ,

so that

(4.3)  δ_{z}^{j}L(z, q) = ∑_{m,n≥1} (m^{j}z^{m}q^{mn} − (−m)^{j}z^{−m}q^{mn}) ,

and

(4.4)  δ_{z}^{j}L(z, q)|_{z=1} = \{ \begin{matrix}0, & j even, \\ 2Φ_{j}, & j odd. \end{matrix}

Assume a is even and apply δ_{z}^{a−1} to both sides of (4.1). Then

(4.5)  δ_{z}^{a}C = ∑_{j} \binom{a − 1}{j}δ_{z}^{j}(L) δ_{z}^{a−1−j}(C).

Setting z = 1 and using (4.4) and (1.36) we obtain the following recurrence

(4.6)  C_{a} = 2 ∑_{j=1}^{\frac{a}{2} −1} \binom{a − 1}{2j − 1} Φ_{2j−1} C_{a−2j} + 2Φ_{a−1} P.

We compute some examples.

(4.7)  C_{2} = 2 P Φ_{1},
C_{4} = 2 P (Φ_{3} + 6 Φ_{1}^{2}) ,
C_{6} = 2 P (Φ_{5} + 30 Φ_{3}Φ_{1} + 60 Φ_{1}^{3}) ,
C_{8} = 2 P (Φ_{7} + 56 Φ_{5}Φ_{1} + 840 Φ_{3}Φ_{1}^{2} + 840 Φ_{1}^{4} + 70 Φ_{3}^{2}) .

Using induction and (4.6) we can show that for n ≥ 1 there are integers α_{a_{1},a_{2},...,α_{n}} such that

(4.8)  C_{2n} = 2 P ∑_{a_{1}+2a_{2}+···+na_{n}=n} α_{a_{1},a_{2},...,α_{n}}Φ_{1}^{a_{1}}Φ_{3}^{a_{2}} ⋯ Φ_{2n−1}^{a_{n}}.

By (2.11) and (4.7) we have

(4.9)  C_{2} = 2δ_{q}P,

or

(4.10)  M_{2}(n) = ∑_{k} k^{2} M(k, n) = 2np(n).

A combinatorial proof of (4.10) was found by Dyson [D2].

We need the following Lemma 4.1. For m ≥ 1, there exists a Φ ∈ W_{m} such that

(4.11)  δ_{q}^{m} (P) = P Φ.

Proof. We proceed by induction on m. From (2.11)

(4.12)  δ_{q}(P) = P Φ_{1},

and the result holds for m = 1. Suppose the result holds for m = a; i.e.,

(4.13)  δ_{q}^{a} (P) = P Φ,

for some Φ ∈ W_{a}. Then

δ_{q}^{a+1} (P) = P δ_{q}(Φ) + δ_{q}(P) Φ,
= P (δ_{q}(Φ) + Φ_{1} Φ),

and the result holds for m = a + 1 since δ_{q}(Φ) ∈ W_{a+1} by (3.29). The result for general m follows by induction. □

We may calculate δ_{q}^{a}(P) by using (4.12) and the recurrence

(4.14)  δ_{q}^{a}(P) = ∑_{j=0}^{a−1} \binom{a − 1}{j} δ_{q}^{j}(Φ_{1}) δ_{q}^{a−j−1}(P),

which is obtained by applying δ_{q}^{a−1} to both sides of (4.12). For example, we have

(4.15)  δ_{q}^{2}(P) = −\frac{1}{6} P (6 Φ_{1}^{2} − 5 Φ_{3} − Φ_{1}) ,
δ_{q}^{3}(P) = \frac{1}{12} P (36 Φ_{1}^{3} − 90 Φ_{1}Φ_{3} − 6 Φ_{1}^{2} + 7 Φ_{5} + 5 Φ_{3}) .

Theorem 4.2. For m ≥ 0 and n ≥ 1, there exists Φ ∈ W_{n+m} such that

(4.16)  δ_{q}^{m} (C_{2n}) = P Φ,

where W_{k} is defined by (3.27).

Proof. Let n ≥ 1. We proceed by induction on m. The result is true for m = 0 using (4.8) and (3.25). The remainder of the proof is analogous to that of Lemma 4.1. □

We give some examples.

δ_{q}(C_{2}) = −\frac{1}{3} P (6 Φ_{1}^{2} − 5 Φ_{3} − Φ_{1}) ,

δ_{q}^{2}(C_{2}) = \frac{1}{6} P (36 Φ_{1}^{3} − 90 Φ_{1}Φ_{3} − 6 Φ_{1}^{2} + 7 Φ_{5} + 5 Φ_{3}) , (4.17)
δ_{q}(C_{4}) = −\frac{1}{15} P (−90 Φ_{1}Φ_{3} − 21 Φ_{5} − 10 Φ_{3} + Φ_{1} + 540 Φ_{1}^{3} − 60 Φ_{1}^{2}) .

5. Relations between rank and crank moments

Let a be even. After applying δ_{z}^{a} to both sides of the rank-crank PDE (2.22), setting z = 1 and using (4.9) we find that

∑_{i=0}^{a/2−1} \binom{a}{2i} ∑_{\begin{matrix}α+β+γ=a−2i \\ α, β, γ even ≥0 \end{matrix}} \binom{a − 2i}{α, β, γ} C_{α} C_{β} C_{γ} P^{−2} − 3 (2^{a−1} − 1) C_{2}

(5.1) = \frac{1}{2}(a − 1)(a − 2)R_{a} + 6 ∑_{i=1}^{a/2−1} \binom{a}{2i} (2^{2i−1} − 1) δ_{q}(R_{a−2i})

+ ∑_{i=1}^{a/2−1} [\binom{a}{2i + 2} (2^{2i+1} − 1) − 2^{2i}\binom{a}{2i + 1} + \binom{a}{2i}] R_{a−2i}.

For a = 2 we obtain 0 = 0. For a = 4 we obtain

(5.2)  C_{4} + 6 \frac{C_{2}^{2}}{P} − C_{2} = R_{4} − R_{2} + 12 δ_{q}(R_{2}).

Using (4.7), (4.17) and (5.2) we find that

(5.3)  −\frac{2}{3} C_{2} − 2 δ_{q}(C_{2}) + \frac{8}{3} C_{4} = R_{4} − R_{2} + 12 δ_{q}(R_{2}),

or

(5.4)  N_{4}(n) = \frac{2}{3} (−3 n − 1) M_{2}(n) + \frac{8}{3} M_{4}(n) + (−12 n + 1) N_{2}(n),

for n ≥ 0.

Similarly, for 2, 3, 4, and 5, there are polynomials P_{k}(n) of degree k − 1 and Q_{k,j}(n) of degree k − j (for 1 ≤ j ≤ k) such that

(5.5)  N_{2k}(n) = P_{k}(n) N_{2}(n) + ∑_{j=1}^{k} Q_{k,j}(n) M_{2j}(n),

for n ≥ 0. For k = 6 there is no such relation. For k = 7 there is a similar relation but with an extra term N_{12}(n). These relations are given in the following

Theorem 5.1. For n ≥ 0 we have

(5.6)  N_{4}(n) = \frac{2}{3} (−3 n − 1) M_{2}(n) + \frac{8}{3} M_{4}(n) + (−12 n + 1) N_{2}(n),

(5.7) N_{6}(n) = \frac{2}{33} (324 n^{2} + 69 n − 10) M_{2}(n) + \frac{20}{33} (−45 n + 4) M_{4}(n)

+ \frac{18}{11} M_{6}(n) + (108 n^{2} − 24 n + 1) N_{2}(n),

(5.8) N_{8}(n) = \frac{2}{913} (−72972 n^{3} − 1728 n^{2} + 5667 n − 289) M_{2}(n)

+ \frac{280}{913} (732 n^{2} − 195 n + 8) M_{4}(n) + \frac{84}{913} (−196 n + 15) M_{6}(n)

+ \frac{1248}{913} M_{8}(n) + (−864 n^{3} + 360 n^{2} − 36 n + 1) N_{2}(n),

N_{10}(n) (5.9)

= \frac{2}{5951847} (3588144480 n^{4} − 805458600 n^{3} − 398007108 n^{2} + 56257647 n − 1794592) M_{2}(n)

+ \frac{140}{5951847} (−72270360 n^{3} + 36826920 n^{2} − 3625245 n + 104002) M_{4}(n)

+ \frac{210}{1983949} (1421544 n^{2} − 380744 n + 13519) M_{6}(n)

+ \frac{120}{1983949} (−282435 n + 18796) M_{8}(n) + \frac{2724}{2173} M_{10}(n)

+ (6480 n^{4} − 4320 n^{3} + 756 n^{2} − 48 n + 1) N_{2}(n)

N_{14}(n) = \frac{1}{4505033323132497} (−655918847016750354240 n^{6} (5.10)

+ 584104439765983424400 n^{5} − 88193910587689930464 n^{4} − 51255985689606317364 n^{3} + 12889219681488512844 n^{2} −1033571808069319887 n + 23432656561492057) M_{2}(n)

+ \frac{364}{4505033323132497} (2544016408481081520 n^{5}

− 2986029950270749200 n^{4} + 1233083592931144500 n^{3} − 185464100558325420 n^{2} + 12124758510318780 n −229618708346923) M_{4}(n)

+ \frac{728}{500559258125833} (−12932704022040180 n^{4}

+ 11781511098477120 n^{3} − 3661921161131415 n^{2} + 234233352768436 n −7334109150929) M_{6}(n)+

\frac{364}{500559258125833} (3327634333443960 n^{3}

− 2184561928177200 n^{2} + 464283118670595 n −12774042869566) M_{8}(n)

+ \frac{2002}{6030834435251} (−758615153688 n^{2} + 404700708960 n

−24122003839) M_{10}(n)

+ \frac{25388554464}{2775349487} (n − 1) M_{12}(n)

+ \frac{139497552}{120667369} M_{14}(n)

+ \frac{1}{138} (−107775360 n^{6} + 143700480 n^{5}

−70752528 n^{4} + 14978304 n^{3} − 1456488 n^{2} + 64320 n − 1045) N_{2}(n)

+ \frac{91}{138} (−36 n + 13) N_{12}(n)

Proof. We first show that relations such as (5.6)–(5.10) must exist. For k ≥ 1 we define

T_{k} = (2k − 1)(k − 1)R_{2k} + 6 ∑_{i=1}^{k−1} \binom{2k}{2i} (2^{2i−1} − 1) δ_{q}(R_{2k−2i})

(5.11)  + ∑_{i=1}^{k−1} [\binom{2k}{2i + 2} (2^{2i+1} − 1) − 2^{2i}\binom{2k}{2i + 1} + \binom{2k}{2i}] R_{2k−2i};

i.e., T_{k} is the function on the right side of (5.1) with a = 2k. By considering the left side of (5.1) and using Theorem 4.2 we have

(5.12)  T_{k} ∈ P W_{k};

i.e., there is a Φ ∈ W_{k} such that

(5.13)  T_{k} = P Φ.

For k ≥ 1 we consider the set of functions

(5.14)  C_{k} = {δ_{q}^{m}(C_{2j}) : 1 ≤ j ≤ k, j + m ≤ k} ⊂ P W_{k}.

Then

(5.15)  |C_{k}| = \frac{k(k + 1)}{2}.

For 1 ≤ k ≤ 5 we observe from Corollary 3.6 that

(5.16)  dim W_{k} = \frac{k(k + 1)}{2}.

Clearly,

(5.17)  dim P W_{k} = dim W_{k},

for all k. Thus, for 1 ≤ k ≤ 5 there is a linear relation between T_{k} and the functions in C_{k}. For k = 1 the relation is T_{1} = 0, which is trivial. For each 2 ≤ k ≤ 5 we have used maple to find this relation. This gives identities (5.6)–(5.9). Corollary 3.6 gives

(5.18)  dim W_{6} = 22.

Since |C_{6}| = 21 there is no reason to expect an analogous relation for k = 6, and in fact calculation shows that the functions in C_{6} together with T_{6} are linearly independent. We note that

(5.19)  T_{6}, δ_{q}(T_{6}), T_{7} ∈ P W_{7}.

Luckily we have

(5.20)  dim W_{7} = 30, and dim C_{7} = 28,

so there must exist a linear relation between the three functions in (5.19) and the functions in C_{7}. We used maple to find this relation and (5.10). □

Although (5.5) does not hold for k = 6. There is a relation for k = 6 but involving an additional function. For an integer r define p_{r}(n) by

(5.21)  ∑_{n≥0} p_{r}(n)q^{n} = ∏_{n=1}^{∞}(1 − q^{n})^{r}.

It is well-known that

(5.22)  Δ(q) = ∑_{n≥1} τ(n)q^{n} = ∑_{n≥1} p_{24}(n − 1)q^{n} = q ∏_{n=1}^{∞}(1 − q^{n})^{24}

is a modular form of weight 12. See [S,pp.95–97]. It follows that

(5.23)  P Δ = ∑_{n≥1} p_{23}(n − 1)q^{n} = q ∏_{n=1}^{∞}(1 − q^{n})^{23} ∈ P W_{6}.

By (5.18) and the fact that |C_{6}| = 21 we see that there must be a linear relation between P Δ, T_{12} and the functions in C_{6}. We used maple to find this relation. It is given in the following

Theorem 5.2. For n ≥ 1 we have

p_{23}(n − 1) = (5.24)
\frac{1}{897196601564928} (−57917897540518785552 n^{5} + 30652078276547889552 n^{4}

+5952274737922797228 n^{3} − 2214892612179680256 n^{2} + 188772676333745691 n

−4410708034409819) M_{2}(n)

+ \frac{5}{224299150391232} (4089872889595634400 n^{4} − 3320629034843596140 n^{3}

+593555423164294752 n^{2} − 40741028214970311 n + 815166233039851) M_{4}(n)

+ \frac{13}{7120607948928} (−4612652508217680 n^{3} + 2500384365901740 n^{2}

−190834728931028 n + 5730847932535) M_{6}(n)

+ \frac{65}{24922127821248} (431597256867684 n^{2} − 112947999359631 n + 3472477850182) M_{8}(n)

+ \frac{143}{600533200512} (−555655003092 n + 33496841951) M_{10}(n) + \frac{16986177}{1919176} M_{12}(n)

+ \frac{24599722121}{3316336128} (−46656 n^{5} + 45360 n^{4} − 12096 n^{3} + 1296 n^{2} − 60 n + 1) N_{2}(n)

− \frac{24599722121}{3316336128} N_{12}(n).

6. Congruence relations between rank and crank moments

In the previous section we considered exact linear relations between rank and crank moments. In this section we consider the analogous problem modulo p. One way to find such relations is to examine the denominators of the rational numbers that occur in an exact relation. For example, if we multiply both sides of (5.7) by 11 and reduce modulo 11 we find

(6.1)  M_{6}(n) ≡ (2n + 3)M_{4}(n) − (n + 8)^{2}M_{2}(n) (mod 11).

We could try the same sort of thing with (5.6). If we multiply both sides of (5.6) by 3 and reduce modulo 3 we obtain

(6.2)  M_{4}(n) ≡ M_{2}(n) (mod 3),

but this relation is trivial since k^{4} ≡ k^{2} (mod 3), for all integers k.

There is another way in which a congruence between rank and crank moments may arise. In (5.14) we defined the set C_{k}. Let k ≥ 1. The set C_{k} seems to be linearly independent over ℚ. For small k and certain primes p the set C_{k} is linearly dependent over ℤ_{p}. For example, consider the case k = 3.

C_{3} = {C_{2}, δ_{q}(C_{2}), δ_{q}^{2}(C_{2}), C_{4}, δ_{q}(C_{4}), C_{6}}.

We want to find congruences between the elements of C_{3}. We consider the 6 × 6 matrix A whose (i, j)-th entry is the coefficient of q^{i} in the q-expansion of the jth element of C_{3}.

A = ( \begin{matrix}M_{2}(1) & M_{2}(1) & M_{2}(1) & M_{4}(1) & M_{4}(1) & M_{6}(1) \\ M_{2}(2) & 2 M_{2}(2) & 4 M_{2}(2) & M_{4}(2) & 2 M_{4}(2) & M_{6}(2) \\ M_{2}(3) & 3 M_{2}(3) & 9 M_{2}(3) & M_{4}(3) & 3 M_{4}(3) & M_{6}(3) \\ M_{2}(4) & 4 M_{2}(4) & 16 M_{2}(4) & M_{4}(4) & 4 M_{4}(4) & M_{6}(4) \\ M_{2}(5) & 5 M_{2}(5) & 25 M_{2}(5) & M_{4}(5) & 5 M_{4}(5) & M_{6}(5) \\ M_{2}(6) & 6 M_{2}(6) & 36 M_{2}(6) & M_{4}(6) & 6 M_{4}(6) & M_{6}(6) \end{matrix} )

= ( \begin{matrix}2 & 2 & 2 & 2 & 2 & 2 \\ 8 & 16 & 32 & 32 & 64 & 128 \\ 18 & 54 & 162 & 162 & 486 & 1458 \\ 40 & 160 & 640 & 544 & 2176 & 8320 \\ 70 & 350 & 1750 & 1414 & 7070 & 32710 \\ 132 & 792 & 4752 & 3300 & 19800 & 103092 \end{matrix} )

We find that

det(A) = −110361968640 = −2^{17} · 3^{7} · 5 · 7 · 11.

Since the determinant is nonzero the functions in C_{3} are linearly independent. If there is a linear congruence for the functions in C_{3} (mod p), then p will divide this determinant. The occurrence of p = 11 is confirmed by (6.1). We suspect that there may be a relation modulo 7. We find that

(6.3)  (n + 2)M_{4}(n) + (6n^{2} + 4n + 1)M_{2}(n) ≡ 0 (mod 7),

for all n. This is easily proved by writing the generating function of the left side of (6.3) in terms of P, Φ_{1}, Φ_{3}, Φ_{5}: (6.4)

(δ_{q} +2)C_{4}+(6δ_{q}^{2}+4δ_{q} +1)C_{2} = −\frac{7}{15} P (180 Φ_{1}Φ_{3} − 30 Φ_{1}^{2} − 18 Φ_{5} − 35 Φ_{3} − 7 Φ_{1})

We have used both methods described above to obtain congruences between rank and crank moments. These are collected together in the following Theorem 6.1. For n ≥ 0 we have

(6.5)  (n + 2)M_{4}(n) + (6n^{2} + 4n + 1)M_{2}(n) ≡ 0 (mod 7),
(n + 5)^{3} M_{4}(n) ≡ (5 n^{4} + 10 n^{3} + 8 n^{2} + 8 n + 9) M_{2}(n) (mod 11), (6.6)
(6.7) M_{6}(n) ≡ 2(n + 7)M_{4}(n) − (n + 8)^{2}M_{2}(n) (mod 11),
(6.8) M_{8}(n) ≡ 2 (n + 5) (n^{2} + 5 n + 10) M_{2}(n) + 6 (n^{2} + n + 1) M_{4}(n) (mod 11),
M_{10}(n) ≡ 4 (n + 7) (n^{3} + 5 n^{2} + 29 n + 32) M_{2}(n) (6.9)

+ 39 (n + 7) (n + 14) (n + 39) M_{4}(n) + (6 n^{2} + 34 n + 39) M_{6}(n)

+ 35 (n + 13) M_{8}(n) (mod 41),
(n + 7) (n + 25) (n + 31) (n^{2} + 28 n + 6) N_{2}(n) + N_{12}(n) (6.10)

≡ (4 n^{5} + 10 n^{4} + 6 n^{3} + 30 n^{2} + 31 n + 33) M_{2}(n) + (n^{4} + 4 n^{3} + 42 n^{2} + 24 n + 30) M_{4}(n) + 40 (n + 10) (n^{2} + 32 n + 7) M_{6}(n) + 31 (n + 13) (n + 41) M_{8}(n) + (n + 11) M_{10}(n) + 22 M_{12}(n) (mod 43),

M_{10}(n) ≡ 36 (n + 19) (n^{3} + 50 n^{2} + 20 n + 36) M_{2}(n) (6.11)

+ (52 n^{3} + 28 n^{2} + 26 n + 52) M_{4}(n) + (36 n^{2} + 11 n + 32) M_{6}(n)

+ 47 (n + 17) M_{8}(n) (mod 53),
M_{8}(n) ≡ (10 n^{3} + 73 n^{2} + 40 n + 82) M_{2}(n) + (72 n^{2} + 23 n + 28) M_{4}(n) (6.12)
+ 10 (n + 41) M_{6}(n) (mod 83),
N_{12}(n) − 367 (n + 332) (n + 487) (n + 664) (n^{2} + 265 n + 155) N_{2}(n) (6.13)

≡ 352 (n + 247) (n + 734) (n^{3} + 147 n^{2} + 597 n + 363) M_{2}(n) + 88 (n + 530) (n + 701) (n + 709) (n + 740) M_{4}(n) + 577 (n + 114) (n + 427) (n + 682) M_{6}(n) + (295 n^{2} + 177 n + 674) M_{8}(n) + 271 (n + 336) M_{10}(n) + 654 M_{12}(n) (mod 797),

M_{14}(n) ≡ (44976165 n^{6} + 23584476 n^{5} + 19728425 n^{4} + 8711555 n^{3} + 36781660 n^{2} (6.14)
+70780973 n + 108798274) M_{2}(n)

+ 77429163 (n + 4141548) (n + 113894720) (n^{3} + 42853554 n^{2} + 28914352 n

+100598975) M_{4}(n)

+ (12571854 n^{4} + 82951807 n^{3} + 9501843 n^{2} + 38248242 n + 118847240) M_{6}(n)

+ 84218605 (n + 53645347) (n^{2} + 6688335 n + 93582728) M_{8}(n)
+ 73449678 (n + 40889782) (n + 59666357) M_{10}(n)
+ 89188917 (n + 120667368) M_{12}(n)
(mod 120667369).

Proof. We have already discussed (6.5) and (6.7). (6.12) follows from multiplying both sides of (5.8) by 83 and reducing modulo 83. (6.6) and (6.11) follow from multiplying both sides of (5.9) by 11 and 53 respectively. (6.14) follows from multiplying both sides of (5.10) by 120667369 and reducing modulo 120667369. The remaining relations follow by writing the generating functions for differences between the left and right sides in terms of P, Φ_{1}, Φ_{3}, and Φ_{5}, and checking that the coefficient of each monomial is divisible by the appropriate prime. □

We may multiply both sides of (5.10) by 23 and then reduce modulo 23 to find a congruence for rank and crank moments. A stronger relation follows from reducing (5.24) modulo 23 and using

(6.15)  ∑_{n≥1} p_{23}(n − 1)q^{n} = q ∏_{n=1}^{∞}(1 − q^{n})^{23} ≡ q ∏_{n=1}^{∞}(1 − q^{23n}) (mod 23),

and Euler’s pentagonal number theorem [An, p.11]

(6.16)  ∏_{n=1}^{∞}(1 − q^{n}) = ∑_{n=−∞}^{∞}(−1)^{n}q^{n(3n+1)/2}.

This gives

Theorem 6.2. For n ≥ 0 we have

4 (n^{2} + n + 14) (n^{3} + n^{2} + 15) M_{2}(n) + (10 n^{4} + 2 n^{3} + 8 n^{2} + 21 n + 22) M_{4}(n)

+ 13 (n + 18) (n^{2} + 21 n + 13) M_{6}(n) + 5 n (n + 6) M_{8}(n) + 15 (n + 19) M_{10}(n) + M_{12}(n) + 12 (n + 10) (n + 14) (n + 19) (n + 20) (n + 21) N_{2}(n) + N_{12}(n)

≡ \{ \begin{matrix}(−1)^{k} (mod 23), & if n = 23k(3k ± 1)/2 + 1, \\ 0 (mod 23), & otherwise. \end{matrix}

Acknowledgement. We would like to thank the referee for alerting us to the paper of Kaneko and Zagier [K-Z].

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Department of Mathematics, University of Illinois at Chicago, 851 S. Morgan 24 A.O.L. ATKIN AND F.G. GARVAN Street, Chicago, Illinois 60607-7045

Department of Mathematics, University of Florida, Gainesville, Florida 32611

   E-mail address: frank@math.ufl.edu