Some congruences for $(s,t)$-regular bipartitions modulo $t$

TL;DR

This paper investigates congruences for the function counting $(s,t)$-regular bipartitions, extending known results by proving new infinite families of congruences modulo various primes for specific $(s,t)$ pairs.

Contribution

It establishes new infinite families of congruences for $B_{s,t}(n)$ for various $(s,t)$, expanding the understanding of bipartition congruences beyond previous results.

Findings

Proves congruences modulo 5 for $B_{2,15}(n)$.

Establishes congruences modulo 11 for $B_{7,11}(n)$ and $B_{27,11}(n)$.

Demonstrates congruences modulo 17 for $B_{243,17}(n)$.

Abstract

In this work, we study the function $B_{s,t}(n)$, which counts the number of $(s,t)$-regular bipartitions of $n$. Recently, many authors proved infinite families of congruences modulo $11$ for $B_{3,11}(n)$, modulo $3$ for $B_{3,s}(n)$ and modulo $5$ for $B_{5,s}(n)$. Very recently, Kathiravan proved several infinite families of congruences modulo $11$, $13$ and $17$ for $B_{5,11}(n)$, $B_{5,13}(n)$ and $B_{81,17}(n)$. In this paper, we will prove infinite families of congruences modulo $5$ for $B_{2,15}(n)$, modulo $11$ for $B_{7,11}(n)$, modulo $11$ for $B_{27,11}(n)$ and modulo $17$ for $B_{243,17}(n)$.

Equations

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

$$\sum _{n=0}^{\infty }{p}_k(n)q^n=(q;q{)}_{\infty }^k,$$

$$f_k:=(q^k;q^k{)}_{\infty }=\prod _{m=1}^{\infty }(1-q^{mk}).$$

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

$$\sum _{n=0}^{\infty }p(7n+5)q^n=7\frac{f_7^3}{f_1^4}+49q\frac{f_7^7}{f_1^8}.$$

$$\sum _{n=0}^{\infty }{b}_{\ell }(n)q^n=\frac{f_{\ell }}{f_1}.$$

$$\sum _{n=0}^{\infty }{B}_{s,t}(n)q^n=\frac{f_s{f}_t}{f_1^2}.$$

$$B_{3,11}\left(3^{\alpha }n+\frac{5\cdot 3^{\alpha -1}-1}{2}\right)\equiv 0(\mathrm{mod}11).$$

$$B_{3,s}\left(p^{2\alpha +1}n+\frac{(1+s)(p^{2\alpha +2}-1)}{24}\right)\equiv 0(\mathrm{mod}3).$$

$$B_{5,13}\left(5^{6m+5}(5n+k)+\frac{5^{6m+5}-2}{3}\right)\equiv 0(\mathrm{mod}13),\text{leavevmode}\text{leavevmode}\text{where}\text{leavevmode}\text{leavevmode}k=1,5.$$

$$B_{2,15}\left(3^{2m+1}n+\frac{7\cdot 3^{2m+1}-5}{8}\right)$$

$$B_{2,15}\left(3^{2m+2}n+\frac{23\cdot 3^{2m+1}-5}{8}\right)$$

$$B_{2,15}\left(3^{2(m+1)+1}n+\frac{13\cdot 3^{2(m+1)}-5}{8}\right)$$

$$B_{7,11}\left(7^{12m}n+\frac{2\cdot 7^{12m}-2}{3}\right)$$

$$B_{7,11}\left(7^{12m+11}(7n+k)+\frac{2\cdot 7^{12m+11}-2}{3}\right)$$

$$B_{27,11}\left(3^{m}n+\frac{5\cdot 3^{m-1}-3}{2}\right)$$

$$B_{243,17}(81n+23)$$

$$B_{243,17}(81n+77)$$

$$f_p\equiv f_1^p(\mathrm{mod}p).$$

$$\frac{f_2^2}{f_1}$$

$$\frac{f_2}{f_1^2}=\frac{f_6^4{f}_9^6}{f_3^8{f}_{18}^3}+2q\frac{f_6^3{f}_9^3}{f_3^7}+4q^2\frac{f_6^2{f}_{18}^3}{f_3^6}.$$

$$f_1^3$$

$$a(q)$$

$$\frac{1}{f_1^3}$$

$$a(q)=\sum _{m,n=-\infty }^{\infty }{q}^{m^2+mn+n^2}.$$

$$\sum _{n=0}^{\infty }{p}_5(7n+3)q^n=10f_1^4{f}_7+49q{f}_7^5.$$

$$\sum _{n=0}^{\infty }{p}_5(n)q^n=f_1^5.$$

$$f_1=f_{49}\left(\frac{B(q^7)}{C(q^7)}-\frac{A(q^7)}{B(q^7)}q-q^2+\frac{C(q^7)}{A(q^7)}{q}^5\right),$$

$$\sum _{n=0}^{\infty }{p}_5(n)q^n=f_{49}^5{\left(\frac{B(q^7)}{C(q^7)}-\frac{A(q^7)}{B(q^7)}q-q^2+\frac{C(q^7)}{A(q^7)}{q}^5\right)}^5.$$

$$\sum _{n=0}^{\infty }{p}_5(7n+3)q^n$$

$$-10\left(\frac{A^3}{B{C}^2}-\frac{B^{3}q}{A^{2}C}-\frac{C^3{q}^2}{A{B}^2}\right)-61q).$$

$$\frac{B^5}{A{C}^4}-\frac{A^5}{B^{4}C}-\frac{C^5{q}^3}{A^{4}B}$$

$$\frac{A{B}^2}{C^3}+\frac{A^{2}Cq}{B^3}-\frac{B{C}^2{q}^2}{A^3}$$

$$\frac{A^3}{B{C}^2}-\frac{B^{3}q}{A^{2}C}-\frac{C^3{q}^2}{A{B}^2}$$

$$\frac{B^7}{C^7}-\frac{A^{7}q}{B^7}+\frac{C^7{q}^5}{A^7}$$

Full text

Some congruences for $(s,t)$-regular bipartitions modulo $t$

**T. Kathiravan111Email: kkathiravan98@gmail.comand K. Srilakshmi222srilakshmi@iisertvm.ac.in **

11footnotemark: 122footnotemark: 2

School of Mathematics,

Indian Institute of Science Education and Research,

Thiruvananthapuram-695 551, Kerala,

India.

11footnotemark: 1

The Institute of Mathematical Sciences,

CIT Campus, Taramani,

Chennai 600113, India

Abstract:

In this work, we study the function $B_{s,t}(n)$, which counts the number of $(s,t)$-regular bipartitions of $n$. Recently, many authors proved infinite families of congruences modulo $11$ for $B_{3,11}(n)$, modulo $3$ for $B_{3,s}(n)$ and modulo $5$ for $B_{5,s}(n)$. Very recently, Kathiravan proved several infinite families of congruences modulo $11$, $13$ and $17$ for $B_{5,11}(n)$, $B_{5,13}(n)$ and $B_{81,17}(n)$. In this paper, we will prove infinite families of congruences modulo $5$ for $B_{2,15}(n)$, modulo $11$ for $B_{7,11}(n)$, modulo $11$ for $B_{27,11}(n)$ and modulo $17$ for $B_{243,17}(n)$ .

2010 Mathematics Subject Classification: 11P83, 05A17.

Keywords: Congruence, Regular Bipartition.

1 Introduction

For $n$ a positive integer, a partition of $n$ is a non-increasing sequence of positive integers whose sum is $n$. The number of partitions of $n$ is denoted by $p(n)$. The generating function for $p(n)$, is given by

[TABLE]

For a nonzero integer $k$, we define the general partition function $p_{k}(n)$ as the coefficient of $q^{n}$ in the expansion of $(q;q)^{k}_{\infty}$. If $k=-1$ we have usual partition function $p(n)$. The generating function for $p_{k}(n)$, is given by

[TABLE]

where as customary, we define

[TABLE]

In [18, 19] Ramanujan obtained the beautiful identities

[TABLE]

and

[TABLE]

Ramanujan [18] give a brief of the identities (1.3), he did not prove the identities (1.4) in [18], but [20] he did give a sketch of his proof of identities (1.4) in his unpublished manuscript of the partition and $\tau$- function. Note that (1.3) and (1.4) immediately yield the congruences $p(5n+4)\equiv 0\pmod{5}$ and $p(7n+5)\equiv 0\pmod{7}.$

Ramanujan partition congruences motivated an investigation of many classes of partitions, such as $\ell$-regular partitions. For a positive integer $\ell$, a partition is said to be $\ell$-regular if none of its parts is divisible by $\ell$. If $b_{\ell}(n)$ denote the number of $\ell$-regular partitions of $n$, then the generating function for $b_{\ell}(n)$, is given by

[TABLE]

In recent years, many authors studied arithmetic properties of $\ell$-regular partitions [5, 6, 7, 9, 10, 12, 22, 23, 25].

Recall that, for a positive integers $s>1$ and $t>1$, a bipartition $(\lambda,\mu)$ of $n$ is a pair of partitions $(\lambda,\mu)$ such that the sum of all the parts equals $n$. A $(s,t)$-regular bipartition of $n$ is a bipartition $(\lambda,\mu)$ of $n$ such that $\lambda$ is a $s$-regular partition and $\mu$ is a $t$-regular partition. If $B_{s,t}(n)$ denote the number of $(s,t)$-regular bipartitions of $n$, then the generating function $B_{s,t}(n)$, is given by

[TABLE]

Recently, Lin [16] proved infinite families of congruence modulo $3$ for $B_{7}(n)$, by using Ramanujan’s two modular equation of degree $7$, and in [17], he proved infinite families of congruence modulo $3$ for $B_{13}(n)$. For more related works, see [14, 21].

Very recently ,[8] Dou proved that, for $n\geq 0$ and $\alpha\geq 2$,

[TABLE]

Adiga and Ranganatha [1] proved infinite families of congruences modulo $3$ for $B_{3,7}(n)$ and Xia and Yao [24] proved several infinite families of congruences modulo $3$ for $B_{3,s}(n)$, modulo $5$ for $B_{5,s}(n)$ and modulo $7$ for $B_{3,7}(n)$. For example, let $s$ be a positive integer and let $p\geq 5$ be a prime, for $n\geq 0$,

[TABLE]

Very Recently, Kathiravan [15] proved several infinite families of congruences modulo $11$, $13$ and $17$ for $B_{5,11}(n)$, $B_{5,13}(n)$ and $B_{81,17}(n)$. For example, for all $n\geq 0$ and $m\geq 0$,

[TABLE]

In this paper, we will prove several infinite families of congruences modulo $5$, $11$, $11$ and $17$ for $B_{2,15}(n)$, $B_{7,11}(n)$, $B_{27,11}(n)$ and $B_{243,17}(n)$. The main results of this paper are as follows,

Theorem 1.1**.**

For all $n\geq 0$ and $m\geq 0$,

[TABLE]

Theorem 1.2**.**

For all $n\geq 0$ and $m\geq 0$,

[TABLE]

Theorem 1.3**.**

For all $n\geq 0$ and $m\geq 4$,

[TABLE]

Theorem 1.4**.**

For all $n\geq 0$,

[TABLE]

2 The identities

In this section, we prove some lemmas to prove our main results. By the binomial theorem for any prime $p$,

[TABLE]

Lemma 2.1**.**

(Berndt [2, p.49]), we have

[TABLE]

Lemma 2.2**.**

(Hirschhorn and Sellers [11]), we have

[TABLE]

Lemma 2.3**.**

(Hirschhorn [13, Eqs. (21.3.1), (22.1.4) and (39.2.8)]), we have

[TABLE]

where

[TABLE]

Lemma 2.4**.**

For $n\geq 0$, we have

[TABLE]

Proof.

Setting $k=5$ in (1.2), we have

[TABLE]

In[2, p. 303, Entry 17(v)], we have

[TABLE]

where $A$, $B$ and $C$ are defined by

$\displaystyle A=A(q):=\frac{f(-q^{3},-q^{4})}{f(-q^{2})}$,   $\displaystyle B=B(q):=\frac{f(-q^{2},-q^{5})}{f(-q^{2})}$ and $\displaystyle C=C:=\frac{f(-q,-q^{6})}{f(-q^{2})}$

Substituting (2.9) into (2.8), we have

[TABLE]

If we extract those terms in which the power of $q$ is congruent to 3 modulo 7, divide by $q^{3}$ and replace $q^{7}$ by $q$, we have

[TABLE]

From [3, P. 174, Entry 31] and [4, Eq. 3.11 and Eq. 3.15] in the terms of $A$, $B$ and $C$, we have

[TABLE]

Substituting (2.13) and (2.14) into (2) and simplifying, we completed the Lemma 2.4. ∎

Lemma 2.5**.**

[4, Theorem 3.2] For $n\geq 0$, we have

[TABLE]

Lemma 2.6**.**

For $n\geq 0$, we have

[TABLE]

Proof.

Setting $k=9$ in (1.2), we have

[TABLE]

Substituting (2.9) into (2.18), we have

[TABLE]

If we extract those terms in which the power of $q$ is congruent to 4 modulo 7, divide by $q^{4}$ and replace $q^{7}$ by $q$, we have

[TABLE]

Rearrange the above equation, we have

[TABLE]

From above follow that

[TABLE]

Substituting (2.12), (2.13), (2.14) and (2.15) into (2), we have

[TABLE]

Simplify the equation (2), we completed the Lemma 2.6. ∎

3 Congruence for $(2,15)$-regular bipartition

Theorem 3.1**.**

For $n\geq 0$, we have

[TABLE]

Proof.

Setting $s=2$ and $t=15$ in (1.6), we have

[TABLE]

Substituting (2.3) into (3.4), we have

[TABLE]

If we extract those terms in which the power of $q$ is congruent to 2 modulo 3, divide by $q^{2}$ and replace $q^{3}$ by $q$, we have

[TABLE]

Substituting (2.2) into (3.6), we have

[TABLE]

If we extract those terms in which the power of $q$ is congruent to 1 modulo 3, divide by $q$ and replace $q^{3}$ by $q$, we have

[TABLE]

Now replacing $q$ by $q^{2}$ in (2.4), we have

[TABLE]

Substituting (3.9) in (3.8) and extract those terms in which the power of $q$ is congruent to 2 modulo 3, divide by $q^{2}$ and replace $q^{3}$ by $q$, we have

[TABLE]

This completed the proof Theorem 3.1, follow from (3.7), (3.10). Substituting (3.9) in (3.8) and extract those terms in which the power of $q$ is congruent to 1 modulo 3, divide by $q$ and replace $q^{3}$ by $q$, we have (3.2). ∎

Proof Theorem 1.1

Equation (1.7) follow from (3.3) by the mathematical induction. Employing (3.1) and (3.2) in (1.7) we obtain (1.8) and (1.9).

4 Congruence for $(7,11)$-regular bipartition

Proof of Theorem 1.2

Setting $s=7$ and $t=11$ in (1.6), we have

[TABLE]

It follows that

[TABLE]

Now from (2.17) and (4.2), we have

[TABLE]

Substituting (2.8), (2.9) and (2.18) into (4), we have

[TABLE]

It follows that

[TABLE]

Substituting Lemma 2.4 and Lemma 2.6 into (4.5), we have

[TABLE]

Similarly, we find

[TABLE]

Substituting (2.9) into (4.7), we have

[TABLE]

There are no terms on the right of $q^{7n+1}$, $q^{7n+5}$ and $q^{7n+6}$, so

[TABLE]

and

[TABLE]

Equation (1.10) follow from (4.10) by mathematical induction. Employing (1.10) in (4.9), we obtain (1.11).

5 Congruence for $(27,11)$-regular bipartition

Proof of Theorem 1.3

Setting $s=27$ and $t=11$, we have

[TABLE]

Substituting (2.4) into (5.1), we have

[TABLE]

It follows that

[TABLE]

Substituting (2.4) and (2.5) into (5.2), we have

[TABLE]

It follows that

[TABLE]

Substituting (2.4), (2.5) and (2.6) into (5.3), we have

[TABLE]

It follows that

[TABLE]

Substituting (2.4) and (2.5) into (5.5), we have

[TABLE]

It follows that

[TABLE]

and

[TABLE]

This completes the proof of Theorem 1.3 follow from (5.7) and (5.8).

6 Congruence for $(243,17)$-regular bipartition

Proof of Theorem 1.4

Setting $s=243$ and $t=17$, we have

[TABLE]

Substituting (2.4) into (6.1), we have

[TABLE]

It follows that

[TABLE]

Substituting (2.4) and (2.5) into (6.2), we have

[TABLE]

If we extract those terms in which the power of $q$ is congruent to 1 modulo 3, divide by $q$ and replace $q^{3}$ by $q$, we have

[TABLE]

Substituting (2.4) and (2.5) into (6.3), we have

[TABLE]

If we extract those terms in which the power of $q$ is congruent to 2 modulo 3, divide by $q$ and replace $q^{3}$ by $q$, we have

[TABLE]

Substituting (2.4) and (2.5) into (6.4), we have

[TABLE]

This completes the proof of Theorem 1.4 follow from (6.5).