SHIFTED AND SHIFTLESS PARTITION IDENTITIES

F. G. GARVAN

Abstract. In 1987, George Andrews considered the following question: For which sets of positive integers S and T is p(S, n) = p(T, n− 1) for all n ≥ 1?, where p(S, n) denotes the number of partitions of n into elements of S. Andrews found two non-trivial examples and in 1989, Kalvade found a further six. We prove a new shifted partition identity using the theory of modular functions. We consider other shifted-type identities and shiftless identities. Let a be a fixed positive integer, and let S, T be distinct sets of positive integers. A shiftless identity has the form: p(S, T) = p(T, n) for all n ≠ a. These other identities arise through certain modular transformations.

1. Introduction

Let S and T be sets of positive integers. Let a be a fixed positive integer. A shifted partition identity has the form

p(S, n) = p(T, n − a), for all n ≥ a.

We will mainly consider the case a = 1. Assume a = 1. If S or T is finite then it is not hard to show that S = T = {1}. Andrews [4] found the following two non-trivial examples:

S = {n : n odd or n ≡ ±4, ±6, ±8, ±10 (mod 32)},
T = {n : n odd or n ≡ ±2, ±8, ±12, ±14 (mod 32)};

and

(1.1)  S = {n : n ≡ ±1, ±4, ±5, ±6, ±7, ±9, ±10, ±11, ±13, ±15,
± 16, ±19 (mod 40)},
T = {n : n ≡ ±1, ±3, ±4, ±5, ±9, ±10, ±11, ±14, ±15, ±16,
± 17, ±19 (mod 40)},

Date: March 1, 2001. 1991 Mathematics Subject Classification. Primary: 11P83; Secondary: 05A17,

11F03.

Key words and phrases. partitions, modular functions, theta functions, eta function, shifted partition identities.

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

In these examples, each S and T is the union of arithmetic progressions modulo M for some M; namely M = 32 and M = 40. In fact, each is a union of 24 such arithmetic progressions. Later, Kalvade [10] found five more identities with M = 42, 48 and 60, each also involving the union of 24 arithmetic progressions. Through a computer search we have found a further 48 identities with M = 40, 42, 46, 48, 54, 56, 60, 62, 66, 70 and 72. All but two of these identities involve unions of 24 arithmetic progressions. The remaining two involve unions of 48 arithmetic progressions. In the present paper, we show how the theory of modular functions may be used to prove certain shifted partition identities.

The generating function for p(S, n) is an infinite product:

∑_{n≥0} p(S, n)q^{m} = ∏_{n∈S} \frac{1}{(1 − q^{n})},

where |q| < 1. For the case a = 1, we can write a shifted partition identity as an equivalent q-series identity:

(1.2)  ∏_{n∈S} \frac{1}{(1 − q^{n})} − q ∏_{n∈T} \frac{1}{(1 − q^{n})} = 1.

There is a simple proof of Andrews’s mod 32 and 40 shifted partition identities using Jacobi’s triple product identity [2, p. 21]:

(1.3) ∑_{n=−∞}^{∞}(−1)^{n}z^{n}q^{n(n−1)/2} = ∏_{n=1}^{∞}(1 − zq^{n−1})(1 − z^{−1}q^{n})(1 − q^{n}).

Let T (z, q) denote the left side. By splitting T (z, q) into its even and odd parts (as a function of z) we easily find that

T (z, q) = T (−z^{2}q, q^{4}) − zT (−z^{−2}q, q^{4}),

or

(1.4)  \frac{T (−z^{2}q, q^{4})}{T (z, q)} − z \frac{T (−z^{−2}q, q^{4})}{T (z, q)} = 1.

The idea is to replace q by q^{m}, and let z = q in (1.4) so that the numerator in each term simplifies to 1 after cancellation in the infinite products. This only occurs in two cases: m = 4 giving Andrews’s mod 32 shifted identity, and m = 5 giving Andrews’s mod 40 shifted identity. Other choices for m do not give shifted partition identities but variants in which the parts from some residue classes must be distinct. These types of identities have been considered by Alladi [1].

When comparing shifted identities, quite often it is helpful to write (1.2) in the form:

∏_{n∈T \S} (1 − q^{n}) − q ∏_{n∈S\T} (1 − q^{n}) = ∏_{n∈S∪T} (1 − q^{n}).

For example, Andrews’s modulus M = 40 result involves

T \ S = {n : n ≡ ±3, ±14, ±17 (mod 40)},
S \ T = {n : n ≡ ±6, ±7, ±13 (mod 40)}.

There is another shifted result with M = 40 that Andrews missed. It involves

T \ S = {n : n ≡ ±2, ±9, ±11, ±12 (mod 40)},
S \ T = {n : n ≡ ±4, ±6, ±7, ±13 (mod 40)}.

We prove this identity using the theory of modular functions in section 4.

By considering the effect of modular transformations on shifted identities we were led to consider shiftless partition identities. Let a be a fixed positive integer, and let S, T be distinct sets of positive integers. A shiftless partition identity has the form:

p(S, T) = p(T, n), for all n ≠ a.

Our simplest example is for modulus M = 40. Let

S = {n : n ≡ ±1, ±2, ±5, ±6, ±7, ±8, ±9, ±11,
± 12, ±13, ±15, ±19 (mod 40)},
T = {n : n ≡ ±1, ±3, ±4, ±5, ±6, ±7, ±8, ±13,
± 14, ±15, ±17, ±19 (mod 40)}.

Then

(1.5)  p(S, n) = p(T, n),  for all n ≠ 2.

This identity follows from our shifted result with M = 40 by applying a modular transformation. The details are given in section 4. Other shiftless identities exist for the moduli M = 42, 46, 48, 54, 56, 60, 62, 66 and 72. In section 5 we present some examples of shifted and shiftless identities. Details for these other identities will be left to a later paper.

2. Computer search

Our new shifted partition identities were found via a computer search. The idea is two consider a finite analogue of (1.2). Let N be an integer N > 2. A pair of sets [S, T] is called a truncated ST -pair O(q^{N}) if

(2.1)  ∏_{n∈S} \frac{1}{(1 − q^{n})} − q ∏_{n∈T} \frac{1}{(1 − q^{n})} = 1 + O(q^{N}),

S ⊂ {1, 2, 3, …, N − 1},

and

T ⊂ {1, 2, 3, …, N − 2}.

It is clear that if S, T give a shifted partition theorem then they also give rise to a truncated ST -pair O(q^{N}), for all N. For N ≥ 3, let T_{N} be the set of truncated ST -pairs O(q^{N}). For a given N, we may find all truncated ST -pairs by a boot-strapping method:

0. Initialize: T_{3} = {[{1}, {1}]}

1. Suppose we are given T_{N}. For each [S, T] ∈ T_{N}, we consider four possible [S′, T′]:

\begin{matrix}a. S′ = S, & T′ = T \\ b. S′ = S ∪ \{N\}, & T′ = T \\ c. S′ = S, & T′ = T ∪ \{N − 1\} \\ d. S′ = S ∪ \{N\}, & T′ = T ∪ \{N − 1\} \end{matrix}

For each of these [S′, T′], if

(2.2)  ∏_{n∈S′} \frac{1}{(1 − q^{n})} − q ∏_{n∈T′} \frac{1}{(1 − q^{n})} = 1 + O(q^{N+1}),

then we include [S′, T′] in T_{N+1}.

In this way we may construct the sets T_{N}. The computation in (2.2) would be done using a computer algebra package like Maple. For small N, we may calculate by hand.

We give some examples.

T_{3} = {[{1}, {1}]}
T_{4} = {[{1, 3}, {1, 2}], [{1}, {1}]}
T_{5} = {[{1, 3, 4}, {1, 2, 3}], [{1, 3}, {1, 2}], [{1, 4}, {1, 3}], [{1}, {1}]}
T_{6} = {[{1, 3, 4, 5}, {1, 2, 3}], [{1, 3, 5}, {1, 2}], [{1, 4, 5}, {1, 3, 4}],
[{1, 4}, {1, 3}], [{1, 5}, {1, 4}], [{1}, {1}]}

We have calculated the T_{N} for N ≤ 37. Let t(N) = |T_{N}|, ie. the number of truncated ST -pairs O(q^{N}). We have the following table:

n  t(n)  n  t(n)
3  1  21  2447
4  2  22  3425
5  4  23  4962
6  6  24  6839
7  11  25  10000
8  15  26  13989
9  26  27  21383
10  41  28  30781
11  67  29  48292
12  96  30  70456
13  138  31  110214
14  197  32  159686
15  300  33  253265
16  431  34  374385
17  636  35  591648
18  893  36  876405
19  1258  37  1354888
20  1723

It appears that t(n) grows exponentially. We do not have enough data to make a real conjecture.

All the known shifted partition identities involve sets of integers S and T that remain unchanged on multiplication by −1 (mod M) for a certain modulus M. Armed with T_{37}, we searched for such shifted partition identities with even modulus up to M = 74.

3. Modular Functions

In this section we set up the necessary theory of modular functions to prove our shifted partition identities.

3.1. Background theory. The necessary background theory of modular functions and modular forms may be found in [16], [17], [11] and [15]. Many of the results that we require are contained in [5], [6] and [7]. Let Γ(1) denote the full modular group and as usual let

Γ_{0}(N) := \{(\begin{matrix}a & b \\ c & d \end{matrix}) ∈ Γ(1) : c ≡ 0 (mod N)\} .

Let Γ be a subgroup of Γ(1) with finite index. For a modular function f on Γ and a cusp ζ the order of f (mod Γ) at ζ is denoted by Ord (f; ζ; Γ) and the invariant order of f at ζ is denoted by ord (f; ζ). We have

(3.1)  Ord (f; ζ; Γ) = κ(Γ; ζ) ord (f; ζ)

where κ(Γ; ζ) denotes the fan width of the cusp ζ (mod Γ), and

(3.2)  ord (f | A; ζ) = ord (f; Aζ),

for A ∈ Γ(1). Here we use the usual stroke operator notation

(f | A)(τ) := f(Aτ).

Any non-zero modular function f must satisfy the valence formula:

(3.3)  ∑_{s∈F} Ord (f; s; Γ) = 0,

where F is a fundamental set of Γ and the sum is taken over s with nonzero order so that this is a finite sum. For s ∈ H the order is interpreted in the usual sense. See [16] for more details. We shall use the valence formula to prove certain modular function identities. This is a standard technique.

3.2. Theta products and eta products. We need results from [7] on transformation formulae and multiplier systems for the Dedekind eta function

(3.4)  η(τ) = exp(πi/12) ∏_{n=1}^{∞}(1 − exp(2πiτ)),

and the theta function

ϑ_{1}(v | τ) = −i ∑_{m=−∞}^{∞}(−1)^{m} exp(πiτ(m + \frac{1}{2})^{2}) exp(2πiv(m + \frac{1}{2})),

where τ ∈ H and v ∈ ℂ.

For (\begin{matrix}a & b \\ c & d \end{matrix}) ∈ Γ(1), η(τ) and ϑ(τ) have well-known transformation for-

mulae

(3.5)  η (\frac{aτ + b}{cτ + d}) = ν_{η}(A) \sqrt{cτ + d} η(τ),

and

ϑ_{1} (\frac{v}{cτ + d} |\frac{aτ + b}{cτ + d}) = ν_{ϑ_{1}}(A) \sqrt{cτ + d} exp(πicv^{2}/(cτ + d)) ϑ_{1}(v | τ).

Here ν_{η}(A), ν_{ϑ_{1}}(A) are explicit 24-th and 8-th roots of unity, respectively. Formulae for these multipliers are given in [7, §2]. We have

(3.6)  ν_{ϑ_{1}}(A) = ν_{η}^{3}(A).

Let q = exp(2πiτ). Let N, ρ be integers, N ≥ 1, ρ ∤ N. We define the theta function θ_{ρ;N}(τ) by

(3.7) θ_{ρ;N}(τ) := ∑_{m=−∞}^{∞}(−1)^{m}q^{\frac{1}{8N} (2Nm+2ρ−N)^{2}}

= q^{\frac{1}{8N} (N−2ρ)^{2}} ∏_{m=1}^{∞}(1 − q^{Nm−ρ})(1 − q^{Nm−(N−ρ)})(1 − q^{Nm}),

by Jacobi’s triple product identity [3, (7.1)]. Our function θ_{ρ;N} corresponds to Biagioli’s [7, (2.8)] f_{N,ρ}. From the definition of θ_{ρ;N} we have

(3.8)  θ_{ρ+N;N} = θ_{−ρ;N} = θ_{ρ;N}.

Biagioli [7, Lemma 2.1] gives the following transformation formula:

θ_{ρ;N}(Aτ) = (−1)^{bρ+⌊ρa/N⌋+⌊ρ/N⌋} exp(ρ^{2}πiab/N) ν_{ϑ_{1}} (^{N}A) \sqrt{cτ + d} θ_{aρ;N}(τ), (3.9)

for A = (\begin{matrix}a & b \\ c & d \end{matrix}) ∈ Γ_{0}(N). Here

NA = (\begin{matrix}a & Nb \\ c/N & d \end{matrix}) .

Similarly, from (3.5) we find that

(3.10)  η(Nτ) | A = ν_{η} (^{N}A) \sqrt{cτ + d} η(Nτ).

For a set of integers ρ = {ρ_{1}, ρ_{2}, …, ρ_{k}}, where each ρ_{j} ∤ N, we define the following theta product

θ_{ρ;N}(τ) := ∏_{j=1}^{k} θ_{ρ_{j};N}(τ).

From (3.9) we have the following transformation formula. For A ∈ Γ_{0}(N), (3.11) θ_{ρ;N}(Aτ) = (−1)^{bL(ρ)+M(ρ,a)} exp(Q(ρ)πiab/N) ν_{ϑ_{1}}^{k}(^{N}A) (cτ+d)^{k/2} θ_{aρ;N}(τ),

where

L(ρ) := ∑_{j} ρ_{j},

M(ρ, x) := ∑_{j}⌊ρ_{j}x/N⌋ + ⌊ρ_{j}/N⌋,

Q(ρ) := ∑_{j} ρ_{j}^{2},

and where aρ = {aρ_{1}, …, aρ_{k}}.

In view of (3.8) we define a relation on ℤ. Let N be a fixed positive integer. For x, y ∈ ℤ we define x ∼_{N} y if either x ≡ y (mod N) or x ≡ −y (mod N). This is an equivalence relation. If x ∼_{N} y then

(3.12)  θ_{x;N} = θ_{y;N},

by (3.8). This equivalence relation extends naturally to certain sets of integers. We consider finite sets of integers whose elements are inequivalent mod ∼_{N}. For any two such sets of integers ρ and σ we define ρ ∼_{N} σ if each element ρ of ρ is equivalent to some element σ of σ modulo ∼_{N} and vice versa. For example,

{3, 9, 12, 15, 18, 21, 39, 45, 48, 51} ∼_{40} {1, 3, 5, 8, 9, 11, 12, 15, 18, 19}

The analogue of (3.12) holds for theta products. Thus, if ρ ∼_{N} σ then

(3.13)  θ_{ρ;N} = θ_{σ;N},

There is a group action of ℤ_{N}^{×} on equivalence classes ρ. For (a, N) = 1, we define a ρ in the natural way by

(3.14)  a ρ = {aρ : ρ ∈ ρ}.

Since (a, N) = 1, the aρ are distinct modulo ∼_{N}. In later sections we will need to compute the orbit of certain sets ρ.

3.3. Orders at cusps. We will need the following lemmas

Lemma 3.1, [7]. If (r, s) = 1, then the fan width of Γ_{0}(N) at \frac{r}{s} is

(3.15)  κ (Γ_{0}(N); \frac{r}{s}) = \frac{N}{(N, s^{2})}.

Lemma 3.2, [9]. Let S_{N} be the set of integer pairs (c, a) satisfying

(0) (1, 0) ∈ S_{N};

(1) c > 1, c | N, 1 ≤ a < c, gcd(c, a) = 1, and;

(2) If (c, a), (c, a′) ∈ S_{N} and a′ ≡ a (mod gcd(c, N/c)), then a = a′. Then the set

\{\frac{a}{c} : (c, a) ∈ S_{N}\}

is a complete set of inequivalent cusps for Γ_{0}(N).

We will need results on the orders at cusps of eta-products and thetafunctions. Newman [14] has found necessary and sufficient conditions under which an eta-product is a modular function on Γ_{0}(N). Ligozat [12] has computed the order of a general eta-product at the cusps of Γ_{0}(N). We need the behavior of the modular form η(Nτ) near each cusp of Γ_{0}(N). The following result follows from [9, Lemma 3.5].

Lemma 3.4, [12]. The order at the cusp s = \frac{b}{c} (assuming (b, c) = 1) of the eta function η(Nτ) is

(3.16)  ord (η(Nτ); s) = \frac{c^{2}}{24N} .

Biagoli [7] has computed the order of the theta function θ_{ρ;N} at any cusp.

Lemma 3.5, [7, Lemma 3.2, p.285]. The order at the cusp s = \frac{b}{c} (assuming (b, c) = 1) of the theta function θ_{ρ;N}(τ) (defined above and assuming ρ ∤ N) is

(3.17)  ord (θ_{ρ;N}(τ); s) = \frac{e^{2}}{2N} (\frac{bρ}{e} − ⌊\frac{bρ}{e}⌋ − \frac{1}{2})^{2} ,

where e = (N, c) and ⌊ ⌋ is the greatest integer function.

4. Proof of a shifted partition identity with M = 40

There are two shifted partition identities with modulus M = 40 and a = 1. The first one is the identity (1.1) found by Andrews. We found the second one via a computer search. Let

(4.1)  S = {n : ±1, ±3, ±4, ±5, ±6, ±7, ±13, ±15, ±16, ±17,
± 18, ±19 (mod 40)},
T = {n : ±1, ±2, ±3, ±5, ±9, ±11, ±12, ±15, ±16, ±17,
± 18, ±19 (mod 40)}.

Then

p(S, n) = p(T, n − 1),

for all n ≥ 1. In this section we prove (4.1) in detail. We show how this identity is equivalent to other shifted partition identities and how it leads to a shiftless partition identity (1.5). The proofs of our other shifted partition identities are analogous. Some of the details are given in the next section.

We identify S with the set of “positive” residue classes mod 40:

S ↔ ρ_{1} := {1, 3, 4, 5, 6, 7, 13, 15, 16, 17, 18, 19},

and T with another set of “positive” residue class mod 40:

T ↔ σ_{1} := {1, 2, 3, 5, 9, 11, 12, 15, 16, 17, 18, 19}.

From (3.4), (3.7), we have

∑_{n≥0} p(S, n)q^{n} = \frac{η^{12}(40τ)}{θ_{ρ_{1};40}(τ)},

and

q ∑_{n≥0} p(T, n)q^{n} = \frac{η^{12}(40τ)}{θ_{σ_{1};40}(τ)}.

Hence, our shifted partition identity is equivalent to showing that

(4.2)  (\frac{1}{θ_{ρ_{1};40}(τ)} − \frac{1}{θ_{σ_{1};40}(τ)}) η^{12}(40τ) = 1.

In view of (3.11) we calculate the set of theta functions θ_{aρ_{1};40} for (a, 40) = 1. This reduces to calculating the orbit of the equivalence class ρ_{1} modulo ∼_{40} under multiplication by the group ℤ_{40}^{×}. A calculation shows that this orbit is

{ρ_{1}, ρ_{3}, ρ_{7}, ρ_{9}},

where

ρ_{3} := {1, 3, 5, 8, 9, 11, 12, 14, 15, 17, 18, 19} ∼_{40} 3ρ_{1},
ρ_{7} := {1, 2, 5, 6, 7, 8, 9, 11, 12, 13, 15, 19} ∼_{40} 7ρ_{1},
ρ_{9} := {2, 3, 4, 5, 7, 9, 11, 13, 14, 15, 16, 17} ∼_{40} 9ρ_{1}.

We may extend the definition of ρ_{j} to all j ∈ ℤ_{40}^{×}, by

ρ_{m} = ρ_{−m}, ρ_{m+20} = ρ_{m}.

In this way, we find that

aρ_{j} ∼_{40} ρ_{aj},

for all a, j ∈ ℤ_{40}^{×}. Similarly, we find that the orbit of the equivalence class σ_{1} modulo ∼_{40} under multiplication by the group ℤ_{40}^{×} is

{σ_{1}, σ_{3}, σ_{7}, σ_{9}},

where

σ_{3} := {3, 4, 5, 6, 7, 8, 9, 11, 13, 14, 15, 17} ∼_{40} 3σ_{1},
σ_{7} := {1, 3, 4, 5, 6, 7, 8, 13, 14, 15, 17, 19} ∼_{40} 7σ_{1},
σ_{9} := {1, 2, 5, 7, 9, 11, 12, 13, 15, 16, 18, 19} ∼_{40} 9σ_{1},

and we may extend the definition of σ_{j} in analogous way so that

aσ_{j} ∼_{40} σ_{aj},

for all a, j ∈ ℤ_{40}^{×}.

We calculate θ_{ρ;40}(Aτ), for A ∈ Γ_{0}(40), and ρ = ρ_{j}, σ_{j}. A calculation shows that

(4.3)  Q(ρ_{j}) ≡ Q(σ_{j}) ≡ 0 (mod 80),
(4.4)  L(ρ_{j}) ≡ L(σ_{j}) ≡ 0 (mod 2),

and

(4.5)  M(ρ_{j}, x) ≡ M(σ_{j}, x) (mod 2),

for j = 1, 3, 7 and 9 and (x, 10) = 1. Assuming (x, 10) = 1,

(4.6)  M(ρ_{1}, x) ≡ 1 (mod 2),

if and only if x ≡ ±3 (mod 20);

(4.7)  M(ρ_{3}, x) ≡ 1 (mod 2),

if and only if x ≢ ±1 (mod 20);

(4.8)  M(ρ_{7}, x) ≡ 1 (mod 2),

if and only if x ≡ ±9 (mod 20); and

(4.9)  M(ρ_{9}, x) ≡ 1 (mod 2),

if and only if x ≡ ±7 (mod 20).

We define four functions:

(4.10)  f_{1}(τ) := (\frac{1}{θ_{ρ_{1}}(τ)} − \frac{1}{θ_{σ_{1}}(τ)}) η^{12}(40τ),
(4.11)  f_{3}(τ) := (\frac{1}{θ_{σ_{3}}(τ)} − \frac{1}{θ_{ρ_{3}}(τ)}) η^{12}(40τ),
(4.12)  f_{7}(τ) := (\frac{1}{θ_{ρ_{7}}(τ)} − \frac{1}{θ_{σ_{7}}(τ)}) η^{12}(40τ),
(4.13)  f_{9}(τ) := (\frac{1}{θ_{ρ_{9}}(τ)} − \frac{1}{θ_{σ_{9}}(τ)}) η^{12}(40τ).

We note that {f_{1}, f_{3}, f_{7}, f_{9}} is the orbit of f_{1} by the group Γ_{0}(40). We have

Theorem 4.1. For A = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Γ_{0}(40),

(4.14)  f_{k}(Aτ) = f_{ak}(τ),

where ±ak is reduced (mod 20).

Proof. The functions f_{3}, f_{7}, and f_{9} are obtained by applying a modular transformation to f_{1}. In fact, from (3.6), (3.10), (3.11) and (4.3)–(4.6) we have

(4.15)  f_{1} | A = f_{a},

for A = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Γ_{0}(40). In the subscript of f_{a} we have reduced ±a (mod 20). For k ∈ \{1, 3, 7, 9\}, choose a fixed matrix A_{k} = ( \begin{matrix}k & ∗ \\ ∗ & ∗ \end{matrix}) ∈ Γ_{0}(40), so that

f_{k} = f_{1} | A_{k}.

For any A = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Γ_{0}(40), and k ∈ \{1, 3, 7, 9\},

f_{k} | A = (f_{1} | A_{k}) | A

= f_{1} | (A_{k}A)
= f_{ak},

since A_{k}A ≡ ( \begin{matrix}ak & ∗ \\ ∗ & ∗ \end{matrix}) (mod 40). □

It is well-known that the theta functions θ_{ρ}, θ_{σ} and the eta function η^{12}(40τ), are meromorphic at all cusps A∞, A ∈ Γ(1). This together with the above theorem gives the

Corollary 4.2. Any symmetric polynomial in f_{1}, f_{3}, f_{7}, f_{9} is a modular function of Γ_{0}(40).

Now we are in a position to complete the proof of our shifted partition identity (4.1). This identity is equivalent to showing that

(4.16)  f_{1} ≡ 1.

We define F (τ) by

(4.17)  F := (f_{1} − 1)(f_{3} − 1)(f_{7} − 1)(f_{9} − 1).

Our result is equivalent to showing that F ≡ 0. Clearly, if (4.16) holds, then F ≡ 0. Conversely, if F ≡ 0, then f_{j} ≡ 0, for some j. But Γ_{0}(40) acts transitively on the {f_{k}}, and so

(4.18)  f_{1} = f_{3} = f_{7} = f_{9} ≡ 1.

Let F be a fundamental set for Γ_{0}(40). By the valence formula (3.3), either F ≡ 0 or

(4.19)  ∑_{s∈F} Ord (F; s; Γ_{0}(40)) = 0.

A set of inequivalent cusps for Γ_{0}(40) is

(4.20)  \{\frac{0}{1}, \frac{1}{40}, \frac{1}{20}, \frac{1}{10}, \frac{1}{8}, \frac{1}{5}, \frac{1}{4}, \frac{1}{2}.\}

From the definition of the f_{j} we find that

ord (f_{j}; s) ≥ min {−ord (θ_{ρ_{1}}; s), −ord (θ_{σ_{1}}; s)} + 12 ord (η(40τ), s).

Using (3.1), (3.15), (3.16) and (3.17) this gives lower bounds for Ord (f_{j}; s; Γ_{0}(40)) at each cusp s of Γ_{0}(40), and j = 1, 3, 7, 9. We denote this lower bound by ℓ(j, s). We have

(4.21)  Ord (f_{j}; s; Γ_{0}(40)) ≥ ℓ(j, s),

where

ℓ(j, 0) = −1, for all j,
ℓ(7, 1/40) = −2,

ℓ(1, 1/20) = −1

and

ℓ(j, s) = 0,

for all other (j, s). Since F has no poles in the complex upper half plane, we have

∑_{s≠i∞} Ord (F; s; Γ_{0}(40)) ≥ −5.

We only need calculate the first couple of terms in the q-expansion of each f_{j}. The lowest power in the q-expansion gives the order at the cusp i∞, which is equivalent to 1/40 mod Γ_{0}(40). Now,

f_{1} = \frac{1}{(1 − q)(1 − q^{3}) ⋯} − q \frac{1}{(1 − q)(1 − q^{2}) ⋯}
= 1 + O(q^{3}),

f_{3} = \frac{1}{(1 − q^{3}) ⋯} − q^{3} \frac{1}{(1 − q) ⋯}
= 1 + O(q^{4}),

f_{7} = \frac{1}{q^{2}(1 − q)(1 − q^{2}) ⋯} − \frac{1}{q^{2}(1 − q)(1 − q^{3}) ⋯}
= O(1),

and

f_{9} = \frac{1}{(1 − q^{2})(1 − q^{3}) ⋯} − q^{2} \frac{1}{(1 − q)(1 − q^{2}) ⋯}
= 1 + O(q^{3}).

Hence,

Ord (F, i∞, Γ_{0}(40)) ≥ 3 + 4 + 0 + 3 = 10,

and

∑_{s∈F} Ord (F; s; Γ_{0}(40)) ≥ 5.

Hence,

F ≡ 0,

and our result follows.

We see that our shifted partition identity (4.1) is equivalent to the four identities given in (4.18). We collect these together into the following

Theorem 4.2 (i) Let S ≡ ± {1, 3, 4, 5, 6, 7, 13, 15, 16, 17, 18, 19} (mod 40), and

T ≡ ± {1, 2, 3, 5, 9, 11, 12, 15, 16, 17, 18, 19} (mod 40). Then p(S, n) = p(T, n − 1), for all n ≠ 0.

(ii) Let S ≡ ± {3, 4, 5, 6, 7, 8, 9, 11, 13, 14, 15, 17} (mod 40), and T ≡ ± {1, 3, 5, 8, 9, 11, 12, 14, 15, 17, 18, 19} (mod 40). Then p(S, n) = p(T, n − 3), for all n ≠ 0.

(iii) Let S ≡ ± {1, 2, 5, 6, 7, 8, 9, 11, 12, 13, 15, 19} (mod 40), and T ≡ ± {1, 3, 4, 5, 6, 7, 8, 13, 14, 15, 17, 19} (mod 40). Then p(S, n) = p(T, n), for all n ≠ 2.

(iv) Let S ≡ ± {2, 3, 4, 5, 7, 9, 11, 13, 14, 15, 16, 17} (mod 40), and T ≡ ± {1, 2, 5, 7, 9, 11, 12, 13, 15, 16, 18, 19} (mod 40). Then p(S, n) = p(T, n − 2), for all n ≠ 0.

Part (i) is our shifted partition identity (4.1). Part (iii) is equivalent to the identity

f_{7} ≡ 1,

and is our shiftless partition identity (1.5).

5. Other shifted and shiftless partition identities

In this section we present some other shifted and shiftless partition identities. Details of proofs of these and other identities we have found will be presented in a later paper.

(i) Let S ≡ ± {1, 4, 5, 7, 8, 9, 11, 12, 13, 15, 16, 19} (mod 42), and T ≡ ± {1, 3, 4, 7, 8, 11, 12, 13, 15, 17, 19, 20} (mod 42). Then p(S, n) = p(T, n − 1), for all n ≠ 0.

(ii) Let S ≡ ± {1, 2, 3, 4, 5, 6, 7, 8, 11, 13, 15, 17} (mod 42), and T ≡ ± {1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 17, 19} (mod 42). Then p(S, n) = p(T, n), for all n ≠ 8.

(iii) Let S ≡ ± {1, 3, 4, 5, 7, 13, 15, 16, 17, 18, 19, 22} (mod 46), and T ≡ ± {1, 2, 3, 7, 11, 12, 15, 16, 17, 19, 20, 21} (mod 46). Then p(S, n) = p(T, n − 1), for all n ≠ 0.

(iv) Let S ≡ ± {1, 2, 3, 5, 6, 9, 10, 11, 13, 14, 17, 21} (mod 46), and T ≡ ± {1, 2, 3, 5, 7, 8, 9, 11, 12, 15, 20, 21} (mod 46). Then p(S, n) = p(T, n), for all n ≠ 6.

(v) Let S ≡ ± {1, 5, 6, 8, 9, 10, 11, 13, 15, 19, 20, 23} (mod 48), and T ≡ ± {1, 4, 5, 7, 9, 10, 15, 17, 18, 19, 20, 23} (mod 48). Then p(S, n) = p(T, n − 1), for all n ≠ 0.

(vi) Let S ≡ ± {1, 2, 3, 4, 5, 6, 11, 13, 19, 20, 21, 23} (mod 48), and T ≡ ± {1, 2, 3, 4, 5, 7, 8, 17, 18, 19, 21, 23} (mod 48). Then p(S, n) = p(T, n), for all n ≠ 6.

(vii) Let S ≡ ± {1, 3, 5, 7, 11, 12, 13, 20, 22, 23, 24, 25} (mod 54), and T ≡ ± {1, 2, 5, 9, 11, 12, 17, 19, 22, 23, 24, 25} (mod 54). Then p(S, n) = p(T, n − 1), for all n ≠ 0.

(viii) Let S ≡ ± {1, 2, 3, 4, 5, 7, 9, 11, 16, 17, 24, 25} (mod 54), and T ≡ ± {1, 2, 3, 4, 5, 7, 9, 12, 13, 20, 23, 25} (mod 54). Then p(S, n) = p(T, n), for all n ≠ 11.

(ix) Let S ≡ ± {1, 3, 4, 5, 7, 18, 21, 22, 23, 24, 25, 27} (mod 56), and T ≡ ± {1, 2, 3, 7, 11, 17, 20, 21, 24, 25, 26, 27} (mod 56). Then p(S, n) = p(T, n − 1), for all n ≠ 0.

(x) Let S ≡ ± {1, 5, 7, 8, 10, 12, 13, 15, 18, 21, 23, 27} (mod 56), and T ≡ ± {2, 3, 5, 7, 8, 13, 15, 20, 21, 22, 23, 25} (mod 56). Then p(S, n) = p(T, n), for all n ≠ 1.

(xi) Let S ≡ ± {1, 7, 8, 10, 11, 13, 14, 17, 19, 20, 23, 29} (mod 60), and T ≡ ± {1, 6, 7, 9, 10, 13, 17, 20, 21, 23, 28, 29} (mod 60). Then p(S, n) = p(T, n − 1), for all n ≠ 0.

(xii) Let S ≡ ± {1, 2, 7, 10, 11, 13, 16, 17, 19, 20, 23, 29} (mod 60), and T ≡ ± {1, 3, 4, 10, 11, 13, 17, 18, 19, 20, 27, 29} (mod 60). Then p(S, n) = p(T, n), for all n ≠ 2.

(xiii) Let S ≡ ± {1, 3, 5, 7, 8, 19, 21, 23, 25, 26, 28, 30} (mod 62), and T ≡ ± {1, 2, 5, 7, 13, 18, 21, 23, 24, 27, 29, 30} (mod 62). Then p(S, n) = p(T, n − 1), for all n ≠ 0.

(xiv) Let S ≡ ± {1, 3, 5, 7, 9, 13, 15, 16, 21, 22, 24, 28} (mod 62), and T ≡ ± {1, 3, 6, 7, 8, 10, 15, 19, 21, 23, 25, 28} (mod 62). Then p(S, n) = p(T, n), for all n ≠ 5.

(xv) Let S ≡ ± {1, 10, 11, 12, 13, 14, 15, 16, 19, 21, 23, 25} (mod 66), and T ≡ ± {1, 9, 10, 11, 12, 13, 14, 15, 25, 29, 31, 32} (mod 66). Then p(S, n) = p(T, n − 1), for all n ≠ 0.

(xvi) Let S ≡ ± {1, 4, 5, 6, 7, 9, 11, 13, 16, 21, 23, 28} (mod 66), and T ≡ ± {1, 4, 5, 6, 7, 9, 11, 14, 16, 17, 27, 29} (mod 66). Then p(S, n) = p(T, n), for all n ≠ 13.

(xvii) Let S ≡ ± {1, 3, 4, 5, 6, 7, 9, 11, 13, 14, 15, 16, 17, 19, 23, 24, 25, 26, 27, 28, 29, 31, 33, 34} (mod 70), and T ≡ ± {1, 2, 3, 5, 8, 9, 11, 12, 13, 14, 15, 17, 18, 19, 21, 22, 23, 25, 27, 28, 29, 31, 32, 33} (mod 70). Then p(S, n) = p(T, n − 1), for all n ≠ 0.

(xix) Let S ≡ ± {1, 3, 5, 7, 8, 26, 28, 29, 30, 31, 33, 35} (mod 72), and T ≡ ± {1, 2, 5, 7, 13, 23, 28, 29, 31, 32, 34, 35} (mod 72). Then p(S, n) = p(T, n − 1), for all n ≠ 0.

(xx) Let S ≡ ± {1, 4, 5, 6, 11, 14, 15, 21, 25, 31, 32, 35} (mod 72), and T ≡ ± {1, 4, 5, 7, 10, 11, 16, 25, 26, 29, 31, 35} (mod 72). Then p(S, n) = p(T, n), for all n ≠ 6.

(xxi) Let S ≡ ± {1, 3, 5, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 23, 25, 27, 29, 31, 32, 33, 34, 35} (mod 72), and T ≡ ± {1, 2, 5, 7, 8, 9, 11, 12, 13, 15, 16, 17, 18, 19, 21, 22, 23, 25, 26, 27, 29, 31, 32, 35} (mod 72). Then p(S, n) = p(T, n − 1), for all n ≠ 0.

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

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