On eight colour partitions

TL;DR

This paper investigates the arithmetic properties of the 8-colour partition function, deriving new Ramanujan-type congruences modulo powers of 2 through explicit generating function formulas.

Contribution

It introduces new Ramanujan-type congruences for the 8-colour partition function and provides explicit formulas for its generating functions.

Findings

Established Ramanujan-type congruences modulo higher powers of 2.

Derived explicit formulas for the generating functions of p_8(n).

Enhanced understanding of the arithmetic properties of 8-colour partitions.

Abstract

In this article, we study the arithmetic properties of the partition function $p_8(n)$, the number of 8-colour partitions of $n$. We prove several Ramanujan type congruences modulo higher powers of 2 for the function $p_8(n)$ by finding explicit formulas for the generating functions.

Equations

$$\sum _{n=0}^{\infty }p(n)q^n=\frac{1}{f_1}.$$

$$f_k:=\prod _{j=0}^{\infty }(1-q^{kj})$$

$$f_1=\sum _{k=-\infty }^{\infty }(-1{)}^k{q}^{\frac{k(3k-1)}{2}}.$$

$$\sum _{n=0}^{\infty }p(5n+4)q^n=5\frac{f_5^5}{f_1^6}.$$

$$p(5^{k}n+{\delta }_k)\equiv 0(\mathrm{mod}{5}^k),$$

$$\sum _{n=0}^{\infty }{p}_3(n)q^n=\frac{1}{f_1^3}.$$

$$p_3\left(3^{2\alpha +1}n+\frac{5\cdot 3^{2\alpha +1}+1}{8}\right)\equiv 0(\mathrm{mod}{3}^{2\alpha +2}),$$

$$\sum _{n=0}^{\infty }{p}_8(n)q^n=\frac{1}{(q;q{)}_{\infty }^8}=\frac{1}{f_1^8}.$$

$$p_8(2n+1)$$

$$p_8\left(2^{2\alpha }n+\frac{2^{2\alpha +1}+1}{3}\right)$$

$$p_8\left(2^{2\alpha +1}n+\frac{7\cdot 2^{2\alpha -1}+1}{3}\right)$$

$$p_8\left(2^{2\alpha +1}n+\frac{5\cdot 2^{2\alpha }+1}{3}\right)$$

$$p_8\left(2^{2\alpha +2}n+\frac{13\cdot 2^{2\alpha -1}+1}{3}\right)$$

$$p_8\left(2^{2\alpha +2}n+\frac{19\cdot 2^{2\alpha -1}+1}{3}\right)$$

$$p_8\left(2^{2\alpha +2}n+\frac{11\cdot 2^{2\alpha }+1}{3}\right)$$

$$p_8\left(2^{2\alpha +3}n+\frac{17\cdot 2^{2\alpha }+1}{3}\right)$$

$$p_8\left(2^{2\alpha +2}n+\frac{2^{2\alpha -1}+1}{3}\right)\equiv 0(\mathrm{mod}{2}^{3\alpha +1}).$$

$$f_1^4=\frac{f_4^{10}}{f_2^2{f}_8^4}-4q\frac{f_2^2{f}_8^4}{f_4^2}$$

$$\frac{1}{f_1^4}=\frac{f_4^{14}}{f_2^{14}{f}_8^4}+4q\frac{f_4^2{f}_8^4}{f_2^{10}}.$$

$$S=\frac{q{f}_4^8}{f_1^8},T=\frac{q^2{f}_4^{24}}{f_2^{24}}.$$

$$S^j=T(S^{j-2}+16S^{j-1}).$$

$$f_1^8=\frac{f_4^{20}}{f_2^4{f}_8^8}+16q^2\frac{f_2^4{f}_8^8}{f_4^4}-8q{f}_4^8.$$

$$\frac{f_2^{24}}{f_1^8{f}_4^8}=\frac{f_4^{20}}{f_2^4{f}_8^8}+16q^2\frac{f_2^4{f}_8^8}{f_4^4}+8q{f}_4^8.$$

$$\frac{1}{S}=\frac{f_4^{12}}{q{f}_2^4{f}_8^8}+16q\frac{f_2^4{f}_8^8}{f_4^{12}}-8$$

$$\frac{1}{S}+16=\frac{f_2^{24}}{q{f}_4^{16}{f}_1^8}.$$

$$\frac{1}{S}\left(\frac{1}{S}+16\right)=\frac{f_2^{24}}{q^2{f}_4^{24}}=\frac{1}{T},$$

$$S=T\left(\frac{1}{S}+16\right),$$

$$S^j=T\left(S^{j-2}+16S^{j-1}\right).$$

$$H\left(\sum _{n=0}^{\infty }{a}_n{q}^n\right)=\sum _{n=0}^{\infty }{a}_{2n}{q}^{2n}$$

$$H\left(\frac{1}{S}\right)=-8.$$

$$H\left(S^j\right)=T\left(H\left(S^{j-2}\right)+16H\left(S^{j-1}\right)\right).$$

$$H(S)$$

$$H(S^2)$$

$$H(S^3)=T\left(H(S)+16H(S^2)\right)=24T^2+2048T^3.$$

Full text

1. Introduction

For any nonnegative integer $n$, let $p(n)$ denote the number of partitions of $n$. The generating function of $p(n)$ is given by

[TABLE]

Here and throughout this paper, we set

[TABLE]

for any positive integer $k$. By Euler’s pentagonal number theorem [1, pp. 10-12], we have

[TABLE]

In 1919, Ramanujan [6] obtained the generating function for $p(5n+4)$ as

[TABLE]

It follows from (1.2) that $p(5n+4)\equiv 0\pmod{5}$. Also, he conjectured that for any integer $k\geq 1,$

[TABLE]

where $\delta_{k}$ is the reciprocal modulo $5^{k}$ of 24, and outlined the proof in [7, pp. 156-177], (also see [3]). In 1938, Watson [8] proved the above conjecture and in 1981, Hirschhorn and Hunt [5] gave a simple proof of the conjecture.

Recently, Hirschhorn [4] studied the number of 3-colour partitions of $n$, denoted by $p_{3}(n)$ and is given by

[TABLE]

He obtained several congruences for $p_{3}(n)$ modulo high powers of $3$, namely, for all $\alpha,n\geq 0$,

[TABLE]

which are analogous to Ramanujan’s congruences for the partition function.

In this paper we study the partition function, $p_{8}(n)$, the number of 8-colour partitions of $n$, which satisfies

[TABLE]

We obtain appropriate generating formulae for $p_{8}(n)$ and deduce several Ramanujan-type congruences modulo high powers of 2. The main results of this paper are as follows.

Theorem 1.1**.**

For all nonnegative integers $n$ and $\alpha\geq 1$, we have

[TABLE]

and if $n$ is not a generalised pentagonal number, then

[TABLE]

We provide some definitions and preliminary results in Section 2. We establish generating functions for $p_{8}(n)$ in Section 3 and prove Theorem 1.1 in Section 4.

2. Preliminaries

In this section, we present some preliminary results which are used in proving our main results.

Lemma 2.1**.**

We have

[TABLE]

and

[TABLE]

Proof.

Lemma 2.1 is an immediate consequence of dissection formulas of Ramanujan, collected in Berndt’s book [2, Entry 25, p. 40]. ∎

Lemma 2.2**.**

Let

[TABLE]

Then, for any $j\geq 1$

[TABLE]

Proof.

Squaring (2.1) on both sides, we deduce that

[TABLE]

Setting $-q$ for $q$ in (2.3) and using the fact that $\prod\limits_{j=0}^{\infty}(1+q^{j})=\dfrac{f_{2}^{3}}{f_{1}f_{4}}$, we obtain

[TABLE]

We can rewrite (2.3) as

[TABLE]

and then (2.4) as

[TABLE]

If follows from the definition of $S$ and last equation that

[TABLE]

which yields

[TABLE]

thus for any $j\geq 1$,

[TABLE]

This completes the proof of Lemma 2.2. ∎

Let us define an operator $H$ as follows:

[TABLE]

and $H(1)=1.$

In view of (2.5), we see that

[TABLE]

Also, from Lemma 2.2

[TABLE]

In particular,

[TABLE]

and

[TABLE]

We define an infinite matrix $M=(m_{j,\,k})_{j,\,k\geq 1}$ as follows:

1. (1)

$m_{1\,,1}=8,\ m_{1,\,k}=0$ for all $k\geq 2$. 2. (2)

$m_{2,\,1}=1,\ m_{2,\,2}=128,\ m_{2,\,k}=0,$ for all $k\geq 3$. 3. (3)

$m_{j,\,1}=0$ for all $j\geq 3$. 4. (4)

$m_{j,\,k}=16m_{j-1,\,k-1}+m_{j-2,\,k-1}$ for all $j\geq 3,\ k\geq 2$.

Using the definition of $M$ and induction, it is easy to prove that

1. (5)

$m_{j\,,k}=0$ for all $k>j$. 2. (6)

$m_{j,\,k}=0$ for all $j>2k$. 3. (7)

$m_{2j,\,j}=1$ for all $j\geq 1$. 4. (8)

$m_{j,\,j}=2^{4j-1}$ for all $j\geq 1$.

We omit the details of the proof. The first eight rows of $M$ are given by

[TABLE]

Lemma 2.3**.**

For any positive integer $j$, we have

[TABLE]

Proof.

The second equality follows from (6). We use mathematical induction to prove the first equality. From (2.7) and (2.8), we see that the above identity holds for $j=1,2$. Suppose that (2.10) holds for some integer $j>2$. From (2.6), we see that

[TABLE]

Using (3),(4) and (8) in the above equation, we arrive at

[TABLE]

This completes the induction. ∎

3. Generating functions

In this section we obtain generating functions for the sequences in Theorem 1.1.

Theorem 3.1**.**

We have

[TABLE]

Proof.

We note that from (2.7),

[TABLE]

or

[TABLE]

This proves (3.1). Rewriting (2.9), we see that

[TABLE]

or

[TABLE]

Thus, (3.2) follows from (3.1) and (3.3). ∎

Let us define another infinite matrix $(x_{\alpha,\,i})_{\alpha,j\geq 1}$ by

1. (9)

$x_{1,\,1}=8,\ x_{1,\,k}=0$ for all $k\geq 2$. 2. (10)

[TABLE]

Using the properties of the numbers $m_{i,\,j}$ and induction, we can easily see that

1. (11)

[TABLE]

and

1. (12)

[TABLE]

The first four rows of $x$ are given by

[TABLE]

Theorem 3.2**.**

For any positive integer $\alpha$,

[TABLE]

and

[TABLE]

Proof.

We prove (3.5) and (3.6) by mathematical induction on $\alpha$. From (3.1) and (3.2), it follows that (3.5) and (3.6) hold for $\alpha=1$.

Suppose that (3.5) holds for some positive integer $\alpha>1$. Applying the operator $H$ to both sides gives

[TABLE]

From Lemma 2.3, we have

[TABLE]

Combining (3.7) and (3.8), we obtain

[TABLE]

Interchanging the order of summantion and using properties (5), (6) and (11) to extend the sums to all positive integers, we have

[TABLE]

which is (3.6). Here last equality follows from properties (10) and (11).

Suppose (3.6) holds for some positive integer $\alpha$. Multiplying the above equation by $q$ and applying the operator $H$ to the resulting expression, we find that

[TABLE]

From Lemma 2.3, we have

[TABLE]

Combine (3.9) and (3.10), interchange the order of the summation and extend sums to all positive integers, we get

[TABLE]

which is (3.5) with $\alpha$ replaced by $\alpha+1$. This completes the proof of (3.5) and (3.6) by induction. ∎

4. Congruences

For a positive integer $n$, let $\vartheta_{2}(n)$ be the highest power of $2$ that divides $n$ and $\vartheta_{2}(0)=+\infty$ by convention.

In this section, we consider the powers of 2 that divide the numbers $m_{j,\,k}$ and $x_{j,\,k}$. Using this information we prove Theorem 1.1.

Lemma 4.1**.**

For any integers $j,k\geq 1$ and $k\leq j\leq 2k$, we have

[TABLE]

Proof.

The proof easily follows from the definition of $m_{j,\,k}$ and properties (5)-(8). ∎

Lemma 4.2**.**

For all positive integers $j$ and $k$, we have

[TABLE]

and

[TABLE]

In both cases, equality holds for $k=1$.

Proof.

We use mathematical induction to prove these results, passing alternatively between (4.2) and (4.3). From (3.4) it follows that (4.2) and (4.3) hold for $j=1,2$ and equality hold in each case for $k=1$.

Assume that (4.2) is true for some positive integer $j$. Consider the case $k=1$ of (4.3).

[TABLE]

By our assumption, $\vartheta_{2}(x_{2j-1,\,1})=3j$ and $\vartheta_{2}(x_{2j-1,\,2})\geq 3j+7$. Hence it follows that $\vartheta_{2}(x_{2j,\,1})=3j+3$. For $k\geq 2$, using the definition of $x_{\alpha,\,j}$ and properties (5) and (6) of the numbers $m_{j,\,k}$, we get

[TABLE]

By Lemma 4.1, $\vartheta_{2}(m_{3i,\,i+k})\geq 4(2k-i)-1$, whenever $i\leq 2k$. Using this fact and the induction hypothesis,

[TABLE]

Combining (4.5) and (4.6), we obtain

[TABLE]

From (4.7), we conclude that if (4.2) is true for some positive integer $j$ then (4.3) is also true for that $j$.

Suppose (4.3) is true for some positive integer $j$. Consider

[TABLE]

By our assumption, $\vartheta_{2}(x_{2j,1})=3(j+1)$ and hence $\vartheta_{2}(x_{2j+1,1})=3(j+1)$. Let $k\geq 2$, by the definition of $x_{\alpha,\,j}$ and properties (5) and (6), we see that

[TABLE]

which is (4.2) with $j$ replaced by $j+1$. Here second inequality of the last expression follows from the Lemma 4.1. ∎

Proof of Theorem 1.1.

Congruence (1.4) follows from (3.1). By the binomial theorem, we can see that for all positive integers $k$ and $m$

[TABLE]

In view of (4.2) and (4.8), we see that

[TABLE]

and

[TABLE]

It follows from (3.5), (4.9) and (4.10) that,

[TABLE]

which implies that

[TABLE]

and

[TABLE]

Congruence (1.5) follows from (4.11). By substituting (2.1) in (4.12) and extracting the terms involving odd and even powers of $q$,

[TABLE]

and

[TABLE]

Congruences (1.6) and (1.9) follow from (4.13). Congruences (1.8) and (1.12) follow from (4.14).

From (4.3), we have

[TABLE]

In view of (3.6) and (4.15),

[TABLE]

Substituting (2.2) in (4.16) and extracting the terms involving odd powers of $q$, we see that

[TABLE]

which implies that

[TABLE]

Congruences (1.7) and (1.10) follow from (4.17). Congruence (1.11) follows from (4.18). ∎