Some congruences for $\ell$-regular partitions with certain restrictions

1.   Introduction

For convenience, throughout this paper, we use the notation

A partition of a nonnegative integer $n$ is a nonincreasing sequence of positive integers whose sum is $n$. We denote by ${\rm pod}(n)$ the number of partitions of $n$ with odd parts distinct (and even parts are unrestricted). The generating function of ${\rm pod}(n)$ is given by

where $\psi (q): = \sum\nolimits_{n = 0}^\infty {{q^{n(n + 1)/2}}}.$

The congruence properties of ${\rm pod}(n)$ were first studied by Hirschhorn and Sellers ^[1] in 2010. They proved that for all $\alpha \ge 0$ and $n \ge 0$,

Using modular forms, Radu and Sellers ^[2] established several congruence relations modulo 5 and 7 for ${\rm pod}(n)$, such as

For more details on ${\rm pod}(n)$, one can refer to ^[3,4,5].

Our goal in this paper is to find congruence properties of the function ${\rm{pod}}_\ell(n)$, which enumerates the partitions of $n$ into non-multiples of $\ell$ in which the odd parts are distinct (and even parts unrestricted). For example ${\rm{pod}}_3(7) = 4$, where the relevant partitions are 7, 5+2, 4+2+1, 2+2+2+1. The generating function of ${\rm{pod}_\ell}(n)$ is given by

Recently, for each $\alpha \ge 0$, Gireesh, Hirschhorn, and Naika ^[6] have obtained the generating function for

where $4{\delta _\alpha } \equiv - 1 \pmod {3^{\alpha}}$ if $\alpha$ is even, and $4{\delta _\alpha } \equiv - 1 \pmod {3^{\alpha+1}}$ if $\alpha$ is odd. Saika ^[7] also obtained the congruence properties for ${\rm{pod}}_3(n)$ and found infnite families of congruences modulo 2 and 3. In addition, Veena and Fathima studied ^[8] the divisibility properties of ${\rm{pod}}_3(n)$ by using the theory of modular forms, for example, for $k \ge 1$,

Similarly, by imposing restrictions on the even parts while the odd parts are unrestricted, we also obtain the $\ell$-regular partitions with distinct odd parts. Let ${\rm{ped}}_\ell(n)$ denote the number of $\ell$-regular partitions of $n$ with even parts distinct. The generating function of ${\rm{ped}}_\ell(n)$ is given by

Drema and Saikia ^[9] found some infinite families of congruences modulo 2 and 4 for ${\rm{ped}}_\ell(n)$ when $\ell$ = 3, 5, 7 and 11. For example, for any prime $p \ge 5$, $\left({\frac{{ - 6}}{p}} \right) = - 1$ and $1 \le r \le p - 1$, then for any $\alpha \ge 0$, they proved the congruence relation

In ^[10], Hemanthkumar, Bharadwaj, and Naika established several congruences modulo 16 and 24 for ${\rm{pod}}_9(n)$. The authors also proved an identity connecting ${\rm{pod}}_9(n)$ and ${\rm{ped}}_9(n)$

In this work, we will continue to study the congruence properties of ${\rm{pod}}_\ell(n)$ and ${\rm{ped}}_\ell(n)$. The main purpose of this paper is to prove the following results.

Theorem 1. For any $n \ge 0$, $\ell$ is even, and we have

where ${\sigma _k}(n) = \sum\nolimits_{d\left| n \right.} {{d^k}}$ is the standard divisor function.

Theorem 2. Let $p_i$ be distinct odd primes. For any $m, n \ge 0$, $t > 0$, we have

where $A = \prod\nolimits_{i = 1}^{t({2^m} - 1)} {{p_i}}$ and $s \in \mathbb{Z}$ satisfies $8\left| {As - ({2^m} - 1)} \right.$.

Theorem 3. For any $n\ge 0$, $\alpha \ge 0$, we have

Meanwhile, let $\ell \ge 1$, if $1 \le m < 8\ell$, ${m \equiv 1}\pmod 8$, and $\left({\frac{m}{\ell}} \right) = - 1$, then we have

Theorem 4. For any $n\ge 0$, the following statements hold:

$\rm(i)$If $\ell > 0$ satisfying $\ell \equiv - 1 \pmod 8$ and $l$ is a prime with $l\left| \ell \right.$, then

$\rm(ii)$If $l$ is a prime satisfying $l \equiv - 1 \pmod 8$, then

2.   Preliminaries

We begin with some background on modular forms to prove our main results.

Let $f(z) = \sum\nolimits_{n = 0}^\infty {a(n){q^n}}$, if $f(z) \in {M_k}({\Gamma _0}(N), \chi _N^{})$ is a modular form. Let $p$ be a prime and the operators ${U_p}$ and ${V_p}$ are defined by

which satisfies the following property:

The Hecke operator ${{T_p}}$ is defined by

Lemma 1. [12, P. 18], If $f(z) = \prod\nolimits_{\delta \left| N \right.} {\eta {{(\delta z)}^{{r_\delta }}}}$ is an eta-quotient with $k = \sum\nolimits_{\delta \left| N \right.} {{r_\delta }/2} {\rm{ }}\in \mathbb{Z}$, with the additional that

then $f(z)$ satisfies

Here, the character $\chi _N^{}$ is defined by $\chi _N^{} (d): = \left({\frac{{{{(- 1)}^k}s}}{d}} \right)$, where $s: = \prod\nolimits_{\delta \left| N \right.} {{\delta ^{{r_\delta }}}}$. Moreover, if $f(z)$ is holomorphic (resp. vanishes) at all of the cusps of ${\Gamma _0}(N)$, then $f(z) \in {M_k}({\Gamma _0}(N), \chi _N^{})$ (resp. ${S_k}({\Gamma _0}(N), \chi _N^{})$.

Lemma 2. [12, P. 18], Let $c$, $d$, and $N$ be positive integers with $d\left| N \right.$ and gcd$(c, d) = 1$. If $f(z)$ is an eta-quotient satisfying the conditions of Lemma 1 for $N$, then the order of vanishing of $f(z)$ at the cusp $c/d$ is

3.   Proofs of the results

Before proving Theorem 1, we state here the following identity:

Proof of Theorem 1. Taking logarithms of relation (1.1), we find that

Using the differential operator $q\frac{d}{{dq}}$ to (3.2) yields

namely,

Consequently, we can deduce the following congruence:

By equating the even terms coefficients, we find that

In particular, if $\ell$ is even, we have

This completes the proof Theorem 1. □

Before we prove Theorem 2, we included here the following lemma, which was proved in a letter from Tate to Serre ^[11]. He proved that the action of the Hecke operators are locally nilpoten of modulo 2. For simplicity, we define ${M_k} : = {M_k}({\rm{S}}{{\rm{L}}_2} \mathbb{Z})$, ${S_k} : = {S_k}({\rm{S}}{{\rm{L}}_2} \mathbb{Z})$.

Lemma 3. If $f(z) \in {M_k} \cap \mathbb{Z}[[q]]$ is a modular form, then there exists a positive integer $t \le \dim {M_k} \le \left[{\frac{k}{{12}}} \right] + 1$ such that for any collection of distinct odd primes ${p_1}, {p_2}, \cdots, {p_t}$,

That is, $f$ has some degree of nilpotency that is bounded by $\left[{\frac{k}{{12}}} \right] + 1$, which also bounds the degree of nilpotency of the image of $f$ under any Hecke operator. Based on work of Tate, Mahlburg ^[13] proved some congruences modulo arbitrary powers of 2 for the coefficients of certain quotients of Eisenstein series. In particular, Boylan ^[14] gave a corollary of Tate's result, which asserts that for any $t \le j$ if $f(z) \in {S_{12j}}\pmod 2$, that is,

By using the Boylan's result and similar techniques that are used by Mahlburg, we prove here Theorem 2.

Proof of Theorem 2. Setting $\ell = 2^m$ in (1.1), we have

Replacing $q$ by $q^8$ in (3.7) and multiplying both sides by ${q^{({2^m} - 1)}}$, we obtain

where $\Delta (z)$ is the Delta-function, which is the unique cusp form of weight 12 on ${\rm SL }_2(\mathbb{Z})$. $\Delta {(z)^{({2^m} - 1)}} \in {S_{12({2^m} - 1)}}\left({{\rm{S}}{{\rm{L}}_2}(\mathbb{Z})} \right)$ is a cusp form of weight $12(2^m-1)$. Hence, by using the Boylan's result, for any ${{2^m} - 1}$ distinct odd primes ${p_1}, {p_2}, \cdots, {p_{{2^m} - 1}}$, we have

Furthermore, it is well known that $\frac{1}{2}\Delta {(z)^{({2^m} - 1)}}\left| {{T_{{p_1}}}\left| {{T_{{p_2}}}} \right.} \right.\left| \cdots \right.\left| {{T_{{p_{{2^m} - 1}}}}} \right.\in {S_{12({2^m} - 1)}}({\rm SL }_2(\mathbb{Z}))$ is also a cusp form of weight ${12({2^m} - 1)}$. Applying Boylan's corollary once more, we choose any $2^m-1$ distinct odd primes ${p_{{2^m}}}, {p_{{2^m} + 1}}, \cdots, {p_{2({2^m} - 1)}}$ coprime to ${p_1}, {p_2}, \cdots, {p_{{2^m} - 1}}$, and we can deduce that

By iterating the above process, for any ${t({2^m} - 1)}$ distinct odd primes ${p_1}, {p_2}, \cdots, {p_{t({2^m} - 1)}}$, we can conclude that

It follows from (3.8) that

Since $(8, p_i) = 1$, setting $A = \prod\nolimits_{i = 1}^{t({2^m} - 1)} {{p_i}}$, there exists a unique integer $s$ such that $8\left| {As - ({2^m} - 1)} \right.$. Replacing $8n+s$ by $n$ in (3.10), we obtain

This completes the proof of Theorem 2. □

In order to prove the remaining theorems, we first establish an explicit formula for the action of the Hecke operator $T_p$ on eta-quotient ${\eta ^3}(z){\eta ^3}(Nz)$ for $N \equiv - 1 \pmod 8$, where $p$ is a prime satisfying $p\left| N \right.$.

Lemma 4. If $N > 0$ with $N \equiv - 1 \pmod 8$, suppose $p$ is a prime dividing $N$, then

Proof. We write

Adding the $U_p$ operator to ${\eta ^3}(8z)$, we obtain

Replacing $k$ by $pk$ in (3.12), we have

Since $N \equiv - 1 \pmod 8$, by Lemmas 1 and 2, we have that ${\eta ^3}(8z){\eta ^3}(8Nz)\in {S_3}({\Gamma _0}(8N), {\chi _{2N}^{}})$ is a cusp form of weight 3. Adding the $U_p$ operator to ${\eta ^3}(8z){\eta ^3}(8Nz)$ and employing (3.13), we obtain

Since $p\left| N \right.$, then $\chi_N (p) = 0$, and adding the $T_p$ operator to ${\eta ^3}(8z){\eta ^3}(8Nz)$, we deduce that

Replacing ${q^8}$ by $q$ in (3.14), we can deduce (3.11). □

To prove Theorem 3, we here need to verify some congruence relations. First, it is easy to check that if $f(z) \in {M_k}({\Gamma _0}(N), {\chi _N})$, then by definitions we have $f(z)\left| {U_p = } \right.f(z)\left| {{T_p}} \right.$, for $p$ is a prime satisfying $p\left| N \right.$. Second, by using Lemma 4, we can show the following congruences:

Based on the above results, now we are able to prove Theorem 3.

Proof of Theorem 3. Setting $\ell = 5$ in (1.1), we have

From [15, P. 60], Hirschhorn and Sellers proved the following 2-dissection of $f_5/f_1$:

Employing (3.18) in (3.17), we obtain

Extracting those terms involving $q^{2n}$ in (3.19) and replacing $q^2$ by $q$, we obtain

Applying (3.18) once more, it follows that

Extracting those terms involving $q^{2n}$ in (3.20) and replacing $q^2$ by $q$, we obtain

Multiplying both sides by ${q^2}f_{15}^3$ in (3.21), we have

Applying the ${U_3}, {U_5}$ operators in sequence and employing (3.15) and (3.16), we find that

which yields

Dividing both sides by $q^2{f_1}^3$ in (3.22) and acting the $U_{15}$ operator on both sides, we deduce that

By induction, we can deduce that for $\alpha \ge 0$,

which is the first part of (1.3). Substituting $q$ by $q^8$ and multiplying both sides by $q$, we obtain

where $\Delta (z) = \prod\nolimits_{n = 1}^\infty {{{(1 - {q^n})}^{24}}}.$ If ${m \equiv 1}\pmod 8$ and $\left({\frac{m}{\ell}} \right) = - 1$, then $8\ell n+m$ cannot be a square. This implies that the coefficients of ${{q^{8\ell n + m}}}$ in the lefthand side of (3.24) must be even. Hence, we have

which is the first part of (1.4). Similarly as in the preceding discussion, we observe that the eta quotient ${\eta ^3}(z){\eta ^3}(7z)\in {S_3}\left({{\Gamma _0}(7), {\chi _7}} \right)$ is a Hecke eigenform. Hnece, we can conclude that for $\alpha \ge 0$, the following congruence holds:

which is the second part of (1.3). Furthermore, for ${m \equiv 1}\pmod 8$ and $\left({\frac{m}{\ell}} \right) = - 1$, we have

This completes the proof of Theorem 3. □

As another application of Lemma 4, we are now ready to prove a congruence relation modulo 2 for ${\rm{ped}}_\ell(n)$.

Proof of Theorem 4. By (1.2), we have

Multiplying both sides by ${q^{({\ell} + 1)/8}}{f_{\ell}^6}$, we have

Since $l \left| {\ell} \right.,$ acting the operator $U_l$ to both sides of (3.25) and using Lemma 4, we have

Consequently,

Replacing $n - (l^2 + {\ell})/8l$ by $n$ in (3.26), we obtain

This proves (1.5).

In particular, when ${\ell} = l$ is a prime, we obtain

Comparing the coefficients of $q^n$ on both sides of (3.27), we obtain

We deduce (1.6). □

4.   Conclusions

In this paper, with the help of modular forms, we investigated on some congruence problems for $\ell$-regular partitions with certain restrictions. In future studies, interested readers may examine whether these methods also can be extended to congruence problems for other types of partition functions.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Conflict of interest

The author declares that he has no conflict of interest.