Some congruences modulo 5 and 25 for overpartition

TL;DR

This paper discovers new Ramanujan-type congruences modulo 5 and 25 for overpartition functions, confirming a conjecture and expanding understanding of overpartition congruences.

Contribution

It introduces two new congruences modulo 5 and proves four conjectured congruences modulo 25 for overpartitions, advancing the theory of partition congruences.

Findings

Two new Ramanujan-type congruences modulo 5 for overpartition

Four congruences modulo 25 confirmed for overpartition

Affirmative answer to Dou and Lin's conjecture

Abstract

We present two new Ramanujan-type congruences modulo 5 for overpartition. We also give an affirmative answer to a conjecture of Dou and Lin, which includes four congruences modulo 25 for overpartition.

Equations

$$3,\overline{3},2+1,\overline{2}+1,2+\overline{1},\overline{2}+\overline{1},1+1+1,\overline{1}+1+1.$$

$$\sum _{n\ge 0}\overline{p}(n)q^n=\frac{(-q;q{)}_{\infty }}{(q;q{)}_{\infty }}=\frac{(q^2;q^2{)}_{\infty }}{(q;q{)}_{\infty }^2},$$

$$(a;q{)}_{\infty }=\prod _{n\ge 0}(1-a{q}^n).$$

$$E_k=E_k(q):=(q^k;q^k{)}_{\infty }.$$

$$\overline{p}(80n+t)\equiv 0(\mathrm{mod}5),$$

$$\overline{p}(80n+8)$$

$$\overline{p}(80n+52)$$

$$\overline{p}(80n+68)$$

$$\overline{p}(80n+72)$$

$$\overline{p}(135n+t)\equiv 0(\mathrm{mod}5),$$

$$\phi (q):=\sum _{n=-\infty }^{\infty }{q}^{n^2}\text{ and }\psi (q):=\sum _{n\ge 0}{q}^{n(n+1)/2}.$$

$$\phi (q)=\frac{E_2^5}{E_1^2{E}_4^2},\phi (-q)=\frac{E_1^2}{E_2},\psi (q)=\frac{E_2^2}{E_1},\psi (-q)=\frac{E_1{E}_4}{E_2}.$$

$$\chi (q)=\frac{E_2^2}{E_1{E}_4},\chi (-q)=\frac{E_1}{E_2},f(-q)=E_1.$$

$$E_k^{25}\equiv E_{5k}^5(\mathrm{mod}25)$$

$$\frac{E_5^2}{E_{10}^4}\equiv \frac{E_1^{26}}{E_2^{28}}+q\frac{E_1^2}{E_2^4}(\mathrm{mod}25).$$

$$\psi (q{)}^2-q\psi (q^5{)}^2=\frac{f(-q^5)\phi (-q^5)}{\chi (-q)},$$

$$\frac{E_2^4}{E_1^2}-q\frac{E_{10}^4}{E_5^2}=\frac{E_2{E}_5^3}{E_1{E}_{10}}.$$

$$\frac{E_5^2}{E_{10}^4}$$

$$\frac{1}{E_5{E}_{20}}\equiv \frac{E_2^{40}}{E_1^{21}{E}_4^{21}}+4q\frac{E_1^3{E}_4^3}{E_2^8}(\mathrm{mod}25).$$

$$\phi (q{)}^2-\phi (q^5{)}^2=4q\chi (q)f(-q^5)f(-q^{20}),$$

$$\frac{E_2^{10}}{E_1^4{E}_4^4}-\frac{E_{10}^{10}}{E_5^4{E}_{20}^4}=4q\frac{E_2^2{E}_5{E}_{20}}{E_1{E}_4}.$$

$$\frac{1}{E_5{E}_{20}}$$

$$\equiv \frac{E_1^4{E}_4^4}{E_2^{10}}\left(\frac{E_2^{50}}{E_1^{25}{E}_4^{25}}+4q\frac{E_2^2}{E_1{E}_4}\right)$$

$$=\frac{E_2^{40}}{E_1^{21}{E}_4^{21}}+4q\frac{E_1^3{E}_4^3}{E_2^8}(\mathrm{mod}25).$$

$$p=p(q):=\frac{\phi (q{)}^2-\phi (q^3{)}^2}{2\phi (q^3{)}^2},$$

$$k=k(q):=\frac{\phi (q^3{)}^3}{\phi (q)}.$$

$$E_1=2^{-\frac{1}{6}}{q}^{-\frac{1}{24}}{p}^{\frac{1}{24}}(1-p{)}^{\frac{1}{2}}(1+p{)}^{\frac{1}{6}}(1+2p{)}^{\frac{1}{8}}(2+p{)}^{\frac{1}{8}}{k}^{\frac{1}{2}},$$

$$E_2=2^{-\frac{1}{3}}{q}^{-\frac{1}{12}}{p}^{\frac{1}{12}}(1-p{)}^{\frac{1}{4}}(1+p{)}^{\frac{1}{12}}(1+2p{)}^{\frac{1}{4}}(2+p{)}^{\frac{1}{4}}{k}^{\frac{1}{2}},$$

$$E_4=2^{-\frac{2}{3}}{q}^{-\frac{1}{6}}{p}^{\frac{1}{6}}(1-p{)}^{\frac{1}{8}}(1+p{)}^{\frac{1}{24}}(1+2p{)}^{\frac{1}{8}}(2+p{)}^{\frac{1}{2}}{k}^{\frac{1}{2}}.$$

$$\frac{E_2^{30}}{E_1{E}_4^{31}}+24q\frac{E_1^7{E}_2^6}{E_4^{15}}+21q^2\frac{E_1^{15}{E}_4}{E_2^{18}}\equiv \frac{E_5^3}{E_{10}^2{E}_{20}^3}(\mathrm{mod}25).$$

$$\frac{E_5^3}{E_{10}^2{E}_{20}^3}=\frac{E_5^2}{E_{10}^4}\cdot E_5{E}_{20}\cdot \frac{E_{10}^2}{E_{20}^4}.$$

$$\left(\frac{E_2^{30}}{E_1{E}_4^{31}}+24q\frac{E_1^7{E}_2^6}{E_4^{15}}+21q^2\frac{E_1^{15}{E}_4}{E_2^{18}}\right)\frac{1}{E_5{E}_{20}}\equiv \frac{E_5^2}{E_{10}^4}\frac{E_{10}^2}{E_{20}^4}(\mathrm{mod}25).$$

$$0$$

Full text

Keywords. Overpartition, Ramanujan-type congruence.

2010MSC. Primary 11P83; Secondary 05A17.

1. Introduction

An overpartition of a positive integer $n$ is a partition of $n$ where the first occurrence of each distinct part may be overlined. For example, $3$ has the following eight overpartitions:

[TABLE]

As usual, we use $\overline{p}(n)$ to denote the number of overpartitions of $n$. We also agree that $\overline{p}(0)=1$. It is well known that the generating function of $\overline{p}(n)$ is

[TABLE]

where we adopt the standard notation

[TABLE]

For convenience, when $k$ is a positive integer, we write

[TABLE]

Similar to the standard partition function $p(n)$, many authors also found various Ramanujan-type identities and congruences for $\overline{p}(n)$. The interested readers may refer to [4, 7, 8, 9, 11]. Recently, Dou and Lin [5] presented the following four congruences for $\overline{p}(n)$:

[TABLE]

where $t=8$, $52$, $68$, and $72$. In a subsequent paper, Hirschhorn [6] obtained simple proofs of Dou and Lin’s congruences.

The main purpose of this paper is to give an elementary proof to a conjecture of Dou and Lin [5].

Theorem 1.1**.**

For $n\geq 0$,

[TABLE]

Furthermore, using a powerful method involving modular forms due to Radu and Sellers [10], we also obtain two new congruences modulo $5$ for $\overline{p}(n)$.

Theorem 1.2**.**

For $n\geq 0$,

[TABLE]

where $t=63$ and $117$.

2. Proof of Theorem 1.1

2.1. Preliminaries

We first introduce some notations of Ramanujan’s theta functions. Let

[TABLE]

We can write $\varphi(q)$, $\varphi(-q)$, $\psi(q)$, and $\psi(-q)$ in terms of some $E_{k}$’s defined in the previous section:

[TABLE]

We also write

[TABLE]

At first, we notice that from the binomial theorem,

[TABLE]

holds for any positive integer $k$. The following results also play an important role in our proof of Theorem 1.1.

Lemma 2.1**.**

[TABLE]

Proof.

Ramanujan [3, p. 365, Eq. (18.1)] asserted that

[TABLE]

that is,

[TABLE]

Hence we have

[TABLE]

∎

Lemma 2.2**.**

[TABLE]

Proof.

Ramanujan [2, p. 258, Entry 9(iii)] also asserted that

[TABLE]

that is,

[TABLE]

Hence we have

[TABLE]

∎

Let

[TABLE]

and

[TABLE]

The following relations are due to Alaca and Williams [1].

Lemma 2.3**.**

[TABLE]

[TABLE]

and

[TABLE]

2.2. Proof of Eqs. (1.1) and (1.4)

We need the following interesting identity.

Lemma 2.4**.**

[TABLE]

Proof.

Note that

[TABLE]

Hence it suffices to show

[TABLE]

Thanks to Lemmas 2.1 and 2.2, we can rewrite it as

[TABLE]

or equivalently,

[TABLE]

since the constant term in the series of $E_{1}^{6}E_{2}^{2}E_{4}^{20}$ is $1\in(\mathbb{Z}/25\mathbb{Z})^{\times}$, and hence $E_{1}^{6}E_{2}^{2}E_{4}^{20}$ is invertible in the ring $\mathbb{Z}/25\mathbb{Z}[[q]]$. According to Lemma 2.3, it becoms

[TABLE]

where

[TABLE]

Hence the identity follows obviously. ∎

Now from [6, Eq. (3.4)], we deduce that

[TABLE]

Note that

[TABLE]

has no terms of the form $q^{5n+3}$ and $q^{5n+4}$, while $\frac{E_{10}^{3}E_{20}^{2}}{E_{5}^{7}}$ is a series of $q^{5}$. Hence

[TABLE]

and

[TABLE]

2.3. Proof of Eqs. (1.2) and (1.3)

This time we need

Lemma 2.5**.**

[TABLE]

Proof.

Note that

[TABLE]

It therefore suffices to show

[TABLE]

Again by Lemmas 2.1 and 2.2, we can rewrite it as

[TABLE]

or equivalently,

[TABLE]

According to Lemma 2.3, it becoms

[TABLE]

where

[TABLE]

The lemma follows immediately. ∎

Similarly we see from [6, Eq. (3.3)] that

[TABLE]

Through a similar argument, we conclude that

[TABLE]

and

[TABLE]

3. Proof of Theorem 1.2

3.1. The method of Radu and Sellers

Let $\Gamma:=\mathrm{SL}_{2}(\mathbb{Z})$. For a positive integer $N$, we define

[TABLE]

and

[TABLE]

Let $M$ be a positive integer. We denote by $R(M)=\{r:r=(r_{\delta_{1}},\ldots,r_{\delta_{k}})\}$ the set of integer sequences indexed by the positive divisors $1=\delta_{1}<\cdots<\delta_{k}=M$ of $M$. For a positive integer $m$, we set $[s]_{m}=s+m\mathbb{Z}$. We also denote by $\mathbb{Z}_{m}^{*}$ the set of all invertible elements in $\mathbb{Z}_{m}$, and by $\mathbb{S}_{m}$ the set of all squares in $\mathbb{Z}_{m}^{*}$. For $t\in\{0,\ldots,m-1\}$, we set

[TABLE]

[TABLE]

and

[TABLE]

where $\gamma=\begin{pmatrix}a&b\\ c&d\end{pmatrix}$, $r\in R(M)$, $r^{\prime}\in R(N)$, and $\kappa=\kappa(m)=\gcd(m^{2}-1,24)$.

Finally let

[TABLE]

for some $r\in R(M)$. Radu and Sellers’ result states as

Lemma 3.1**.**

Let $u$ be a positive integer, $(m,M,N,t,r=(r_{\delta}))\in\Delta^{*}$, the set of tuples satisfying conditions given in [10, p. 2255], $r^{\prime}=(r^{\prime}_{\delta})\in R(N)$, $n$ be the number of double cosets in $\Gamma_{0}(N)\backslash\Gamma/\Gamma_{\infty}$, and $\{\gamma_{1},\ldots,\gamma_{n}\}$ $\subset\Gamma$ be a complete set of representatives of the double coset $\Gamma_{0}(N)\backslash\Gamma/\Gamma_{\infty}$. Assume that $p_{m,r}(\gamma_{i})+p_{r^{\prime}}^{*}(\gamma_{i})\geq 0$ for all $i=1,\ldots,n$. Let $t_{\min}:=\min_{t^{\prime}\in P_{m,r}(t)}t^{\prime}$ and

[TABLE]

Then if

[TABLE]

for all $t^{\prime}\in P_{m,r}(t)$, then

[TABLE]

for all $t^{\prime}\in P_{m,r}(t)$.

3.2. Proof of Theorem 1.2

To apply the method of Radu and Sellers, we notice that

[TABLE]

Then we may take

[TABLE]

By the definition of $P_{m,r}(t)$, we have

[TABLE]

Now we can choose

[TABLE]

Let

[TABLE]

It follows by [10, Lemma 2.6] that $\{\gamma_{\delta}:\delta\mid N\}$ contains a complete set of representatives of the double coset $\Gamma_{0}(N)\backslash\Gamma/\Gamma_{\infty}$. One may verify that all assumptions of Lemma 3.1 are satisfied. Hence we obtain the upper bound $\lfloor v\rfloor=37$. Theorem 1.2 follows immediately by checking the two congruences for $n$ from [math] to $37$.

3.3. Remarks

We remark that this method can also be applied to prove Theorem 1.1. At first, note that

[TABLE]

Now for $\overline{p}(80n+8)$ and $\overline{p}(80n+72)$, we choose

[TABLE]

and compute $P_{m,r}(t)=\{8,72\}$. Then we set

[TABLE]

and therefore obtain the upper bound $\lfloor v\rfloor=71$.

Similarly, for $\overline{p}(80n+52)$ and $\overline{p}(80n+68)$, we have

[TABLE]

[TABLE]

and thus the upper bound $\lfloor v\rfloor=71$.

Finally, we see that Theorem 1.1 follows directly from a simple verification.

Acknowledgements

We thank George E. Andrews and Yucheng Liu for some helpful discussions.