Arithmetic properties of (2, 3)-regular overcubic bipartitions

Abstract

Purpose

Let b¯2,3(n), which enumerates the number of (2, 3)-regular overcubic bipartition of n. The purpose of the paper is to describe some congruences modulo 8 for b¯2,3(n). For example, for each α ≥ 0 and n ≥ 1, b¯2,3(8n+5)≡0(mod8), b¯2,3(2⋅3α+3n+4⋅3α+2)≡0(mod8).

Design/methodology/approach

H.C. Chan has studied the congruence properties of cubic partition function a(n), which is defined by ∑n=0∞a(n)qn=1(q;q)∞(q2;q2)∞.

Findings

To establish several congruence modulo 8 for b¯2,3(n), here the author keeps to the classical spirit of q-series techniques in the proofs.

Originality/value

The results established in the work are extension to those proved in ℓ-regular cubic partition pairs.

Citation

Nayaka, S.S. (2023), "Arithmetic properties of (2, 3)-regular overcubic bipartitions", Arab Journal of Mathematical Sciences, Vol. 29 No. 1, pp. 119-129. https://doi.org/10.1108/AJMS-07-2021-0162

1. Introduction

A partition λ of a natural number n is a finite non-increasing sequence of positive integer parts λ_i (1 ≤ i ≤ m) such that

n=λ1+λ2+λ3+⋯+λm.

In this case, we write |λ| = n. The number of partitions of n is denoted by p(n) and the generating function is given by as follows:

∑n=0∞p(n)qn=1(q;q)∞.

Ramanujan’s three famous congruences of p(n) are as follows:

p(5n+4)≡0(mod5),p(7n+5)≡0(mod7),p(11n+6)≡0(mod11).

In [1–3], H.C. Chan has studied the congruence properties of cubic partition function a(n), which is defined by as follows:

∑n=0∞a(n)qn=1(q;q)∞(q2;q2)∞.

B. Kim [4] studied its overpartition analog, the overcubic partition function a¯(n), which is defined by as follows:

∑n=0∞a¯(n)qn=(−q;q)∞(−q2;q2)∞(q;q)∞(q2;q2)∞.

In [5], M.D. Hirschhorn obtained the results satisfied by a¯(n), which appeared in Kim’s paper [4], and Sellers [6] has proved a number of arithmetic properties of a¯(n) by employing elementary generating function methods. Zhao and Zhong [7] studied cubic partition pairs, which are denoted by b(n), and the generating function is as follows:

∑n=0∞b(n)qn=1(q;q)∞2(q2;q2)∞2.

Recently, Kim [8] studied congruence properties of b¯(n), which denotes overcubic partition pairs of n, whose generating function is given by as follows:

∑n=0∞b¯(n)qn=(−q;q)∞2(−q2;q2)∞2(q;q)∞2(q2;q2)∞2.

More recently, Lin [9] studied various arithmetic properties of b¯(n) modulo 3 and 5. For example, for any α ≥ 2, n ≥ 0,

b¯3α(3n+2)≡0(mod3),

for α ≥ 0,

b¯380⋅5α≡0(mod3).

In [10], Naika and Nayaka have established some congruences for ℓ-regular cubic partition pairs. Let b¯2,3(n) denote the number of (2, 3)-regular overcubic bipartitions of n, whose generating function is given by as follows:

(1.1)∑n=0∞b¯2,3(n)qn=(q2;q2)∞2(q3;q3)∞4(q4;q4)∞4(q24;q24)∞2(q;q)∞4(q6;q6)∞2(q8;q8)∞2(q12;q12)∞4.

In this paper, we establish several congruences modulo 8 for b¯2,3(n). These results can be found in Theorems (3.1), and we keep to the classical spirit of q-series techniques in our proofs.

2. Preliminaries

For |ab| < 1, Ramanujan’s general theta function f(a, b) is defined as follows:

f(a,b)≔∑n=−∞∞an(n+1)/2bn(n−1)/2.

Some special cases of f(a, b) are as follows:

φ(q)≔f(q,q)=∑n=−∞∞qn2=(−q;q2)∞2(q2;q2)∞,

ψ(q)≔f(q,q3)=∑n=0∞qn(n+1)/2=(q2;q2)∞(q;q2)∞

and

f(−q)≔f(−q,−q2)=∑n=−∞∞(−1)nqn(3n−1)/2=(q;q)∞.

Where the product representation of f(a, b) arises from Jacobi’s triple product identity [11, p. 35, Entry 19] as follows:

f(a,b)=(−a;ab)∞(−b;ab)∞(ab;ab)∞.

The following dissection formulas to prove our main results.

Lemma 2.1.

For each prime p and n ≥ 1,

(2.1) (q;q)∞pn≡(qp;qp)∞pn−1(modpn).

Lemma 2.2.

The following 2-dissections holds:

(2.2) (q;q)∞2=(q2;q2)∞(q8;q8)∞5(q4;q4)∞2(q16;q16)∞2−2q(q2;q2)∞(q16;q16)∞2(q8;q8)∞,

(2.3) 1(q;q)∞2=(q8;q8)∞5(q2;q2)∞5(q16;q16)∞2+2q(q4;q4)∞2(q16;q16)∞2(q2;q2)∞5(q8;q8)∞.

Lemma (2.2) is a consequence of dissection formulas of Ramanujan, which is collected in Berndt’s book [11, p. 40, Entry 25].

Lemma 2.3.

The following 2-dissections holds:

(2.4) (q3;q3)∞3(q;q)∞=(q4;q4)∞3(q6;q6)∞2(q2;q2)∞2(q12;q12)∞+q(q12;q12)∞3(q4;q4)∞,

(2.5) (q;q)∞3(q3;q3)∞=(q4;q4)∞3(q12;q12)∞−3q(q2;q2)∞2(q12;q12)∞3(q4;q4)∞(q6;q6)∞2,

(2.6) (q3;q3)∞(q;q)∞3=(q4;q4)∞6(q6;q6)∞3(q2;q2)∞9(q12;q12)∞2+3q(q4;q4)∞2(q6;q6)∞(q12;q12)∞2(q2;q2)∞7.

Hirschhorn, Garvan and Borwein [12] proved (2.4) and (2.5). For proof of (2.6), see [13].

Lemma 2.4.

The following 2-dissections holds:

1(q;q)∞(q3;q3)∞=(q8;q8)∞2(q12;q12)∞5(q2;q2)∞2(q4;q4)∞(q6;q6)∞4(q24;q24)∞2

(2.7) +q(q4;q4)∞5(q24;q24)∞2(q2;q2)∞4(q6;q6)∞2(q8;q8)∞2(q12;q12)∞

(q;q)∞(q3;q3)∞=(q2;q2)∞(q8;q8)∞2(q12;q12)∞4(q4;q4)∞2(q6;q6)∞(q24;q24)∞2

(2.8) −q(q4;q4)∞4(q6;q6)∞(q24;q24)∞2(q2;q2)∞(q8;q8)∞2(q12;q12)∞2.

Eqn (2.7) was proved by Baruah and Ojah [14]. Replacing q by − q in (2.7) and using the fact that (−q;−q)∞=f23f1f4, we get (2.8).

Lemma 2.5.

The following 3-dissection hold:

(2.9)(q;q)∞(q2;q2)∞=(q6;q6)∞(q9;q9)∞4(q3;q3)∞(q18;q18)∞2−q(q9;q9)∞(q18;q18)∞−2q2(q3;q3)∞(q18;q18)∞4(q6;q6)∞(q9;q9)∞2.

One can see this identity in [15].

Lemma 2.6.

[11, p. 345, Entry 1 (iv)]. We have the following 3-dissection

(2.10)(q;q)∞3=(q9;q9)∞3(ζ−1−3q+4q3ζ2),

where

(2.11)ζ=(q3;q3)∞(q18;q18)∞3(q6;q6)∞(q9;q9)∞3.

3. Congruences modulo 8 for b¯2,3(n)

Theorem 3.1.

For each α ≥ 0 and n ≥ 1, we have

(3.1) b¯2,3(6n+5)≡0(mod8),

(3.2) b¯2,3(8n+5)≡0(mod8),

(3.3) b¯2,3(12n+7)≡0(mod8),

(3.4) b¯2,3(18n+15)≡0(mod8),

(3.5) b¯2,3(36n+21)≡0(mod8),

(3.6) b¯2,3(72n+38)≡0(mod8),

(3.7) b¯2,3(72n+51)≡0(mod8),

(3.8) b¯2,3(72n+57)≡0(mod8),

(3.9) b¯2,3(216n+99)≡0(mod8),

(3.10) b¯2,3(216n+171)≡0(mod8),

(3.11) b¯2,3(8⋅9α+2n+57⋅9α+1)≡0(mod8),

(3.12) b¯2,3(8⋅9α+2n+35⋅9α+2)≡0(mod8),

(3.13) b¯2,3(2⋅3α+3n+4⋅3α+2)≡0(mod8),

(3.14) b¯2,3(72n+3)≡b2,3(24n+1)(mod8),

(3.15) b¯2,3(36n+3)≡b2,3(12n+1)(mod8).

Proof.

Employing (2.4) and (2.5) in (1.1), we have

∑n=0∞b¯2,3(n)qn=(q4;q4)∞13(q6;q6)∞3(q24;q24)∞2(q2;q2)∞9(q8;q8)∞2(q12;q12)∞7

(3.16) +4q(q4;q4)∞9(q6;q6)∞(q24;q24)∞2(q2;q2)∞7(q8;q8)∞2(q12;q12)∞3

(3.17) +3q2(q4;q4)∞5(q12;q12)∞(q24;q24)∞2(q2;q2)∞5(q6;q6)∞(q8;q8)∞2,

which implies the generating function as follows:

(3.18) ∑n=0∞b¯2,3(2n+1)qn=4(q2;q2)∞9(q3;q3)∞(q12;q12)∞2(q;q)∞7(q4;q4)∞2(q6;q6)∞3.

Invoking (2.1) in (3.18), we obtain the generating function as follows:

(3.19) ∑n=0∞b¯2,3(2n+1)qn≡4(q;q)∞(q3;q3)∞(q2;q2)∞(q6;q6)∞(mod8).

Substituting (2.8) into (3.19), we get the generating function as follows:

(3.20) ∑n=0∞b¯2,3(2n+1)qn≡4(q2;q2)∞2(q8;q8)∞2(q12;q12)∞4(q4;q4)∞2(q24;q24)∞2+4q(q6;q6)∞2(q4;q4)∞4(q24;q24)∞2(q8;q8)∞2(q12;q12)∞2(mod8).

Extracting the terms in which powers of q are congruent to 1 modulo 2 from (3.20), we have the generating function as follows:

(3.21) ∑n=0∞b¯2,3(4n+3)qn≡4(q3;q3)∞2(q2;q2)∞4(q12;q12)∞2(q4;q4)∞2(q6;q6)∞2(mod8).

Invoking (2.1) in (3.21), we obtain as follows:

(3.22) ∑n=0∞b¯2,3(4n+3)qn≡4(q3;q3)∞2(q6;q6)∞2(mod8).

Extracting the terms involving q³ⁿ from (3.22), replacing q³ by q, we have the generating function as follows:

(3.23) ∑n=0∞b¯2,3(12n+3)qn≡4(q;q)∞2(q2;q2)∞2(mod8).

Employing (2.2) into (3.23), we find the generating function as follows:

(3.24) ∑n=0∞b¯2,3(12n+3)qn≡4(q2;q2)∞3(q8;q8)∞5(q4;q4)∞2(q16;q16)∞2(mod8).

Extracting the terms involving q²ⁿ from (3.24), replacing q² by q, we have the generating function as follows:

(3.25) ∑n=0∞b¯2,3(24n+3)qn≡4(q;q)∞3(q4;q4)∞5(q2;q2)∞2(q8;q8)∞2(mod8).

Invoking (2.1) in (3.25), we get the generating function as follows:

(3.26) ∑n=0∞b¯2,3(24n+3)qn≡4(q2;q2)∞2(q;q)∞(mod8).

Ramanujan recorded the following identity in his third note book; for proof, one can see [11, p. 49].

(3.27) ψ(q)=(q2;q2)∞2(q;q)∞=(q6;q6)∞(q9;q9)∞2(q3;q3)∞(q18;q18)∞+q(q18;q18)∞2(q9;q9)∞.

Substituting (3.27) into (3.26), we obtain the generating function as follows

(3.28) ∑n=0∞b¯2,3(24n+3)qn≡4(q6;q6)∞(q9;q9)∞2(q3;q3)∞(q18;q18)∞+4q(q18;q18)∞2(q9;q9)∞(mod8).

Congruence (3.7) follows from (3.28).

Extracting the terms in which powers of q are congruent to 1 modulo 3 from (3.28), we have the generating function as follows:

(3.29) ∑n=0∞b¯2,3(72n+27)qn≡4(q6;q6)∞2(q3;q3)∞(mod8).

The results (3.9) and (3.10) follow from (3.29).

From (3.29), we obtain the generating function as follows:

(3.30) ∑n=0∞b¯2,3(216n+27)qn≡4(q2;q2)∞2(q;q)∞(mod8).

Using the congruences (3.30) and (3.26), we can see that

b¯2,3(216n+27)≡b¯2,3(24n+3)(mod8).

By mathematical induction on α, we find that

(3.31) b¯2,3216⋅9αn+27⋅9α≡b¯2,3(24n+3)(mod8).

Using (3.7) in (3.31), we get (3.12).

Extracting the terms involving q³ⁿ from (3.28), replacing q³ by q, we have the generating function as follows:

(3.32) ∑n=0∞b¯2,3(72n+3)qn≡4(q2;q2)∞(q3;q3)∞2(q;q)∞(q6;q6)∞(mod8).

Invoking (2.1) in (3.32), we get the generating function as follows:

(3.33) ∑n=0∞b¯2,3(72n+3)qn≡4(q;q)∞(mod8).

From (3.20), we can see that

(3.34) ∑n=0∞b¯2,3(4n+1)qn≡4(q;q)∞2(q4;q4)∞2(q6;q6)∞4(q2;q2)∞2(q12;q12)∞2(mod8).

Invoking (2.1) in (3.34), we have the generating function as follows:

(3.35) ∑n=0∞b¯2,3(4n+1)qn≡4(q2;q2)∞(q4;q4)∞(mod8).

Congruence (3.2) follows from (3.34).

Extracting the terms involving q²ⁿ from (3.35), replacing q² by q, we have the generating function as follows:

(3.36) ∑n=0∞b¯2,3(8n+1)qn≡4(q;q)∞(q2;q2)∞(mod8).

Employing (2.9) into (3.36), we obtain the generating function as follows:

(3.37) ∑n=0∞b¯2,3(8n+1)qn≡4(q6;q6)∞(q9;q9)∞4(q3;q3)∞(q18;q18)∞2+4q(q9;q9)∞(q18;q18)∞(mod8).

Extracting the terms in which powers of q are congruent to 1 modulo 3 from (3.37), we have the generating function as follows:

(3.38) ∑n=0∞b¯2,3(24n+9)qn≡4(q3;q3)∞(q6;q6)∞(mod8).

The results (3.6) and (3.8) follow from (3.38).

From (3.38), we have the generating function as follows:

(3.39) ∑n=0∞b¯2,3(24n+9)qn≡4(q;q)∞(q2;q2)∞(mod8).

Using the congruences (3.39) and (3.36), we can see that

b¯2,3(72n+9)≡b¯2,3(8n+1)(mod8).

By mathematical induction on α, we obtain the generating function as follows:

(3.40) b¯2,38⋅9α+1n+9α+1≡b¯2,3(8n+1)(mod8).

Using (3.8) in (3.40), we get (3.11).

Extracting the terms involving q³ⁿ from (3.37), replacing q³ by q, we have the generating function as follows:

(3.41) ∑n=0∞b¯2,3(24n+1)qn≡4(q2;q2)∞(q3;q3)∞4(q;q)∞(q6;q6)∞2(mod8).

Invoking (2.1) in (3.41), we get the generating function as follows:

(3.42) ∑n=0∞b¯2,3(24n+1)qn≡4(q;q)∞(mod8).

Using the congruences (3.33) and (3.42), we obtain (3.14).

From (3.19), it can be rewritten as follows:

(3.43) ∑n=0∞b¯2,3(2n+1)qn≡4(q;q)∞(q2;q2)∞(q3;q3)∞3(mod8).

Employing (2.9) into (3.43), we obtain the generating function as follows:

(3.44) ∑n=0∞b¯2,3(2n+1)qn≡4(q3;q3)∞2(q6;q6)∞(q9;q9)∞4(q18;q18)∞2+4q(q3;q3)∞3(q9;q9)∞(q18;q18)∞(mod8).

Congruence (3.1) follows from (3.44).

Extracting the terms in which powers of q are congruent to 1 modulo 3 from (3.44), we have the generating function as follows:

(3.45) ∑n=0∞b¯2,3(6n+3)qn≡4(q;q)∞3(q3;q3)∞(q6;q6)∞(mod8),

which implies as follows:

(3.46) ∑n=0∞b¯2,3(6n+3)qn≡4(q;q)∞3(q3;q3)∞3(mod8).

Substituting (2.10) into (3.46), we obtain the generating function as follows:

(3.47) ∑n=0∞b¯2,3(6n+3)qn≡4(q3;q3)∞2(q6;q6)∞(q9;q9)∞6(q18;q18)∞3+4q(q3;q3)∞3(q9;q9)∞3(mod8).

Congruence (3.4) follows from (3.47).

Extracting the terms in which powers of q are congruent to 1 modulo 3 from (3.47), we get the generating function as follows:

(3.48) ∑n=0∞b¯2,3(18n+9)qn≡4(q;q)∞3(q3;q3)∞3(mod8).

Using the congruences (3.48) and (3.46), we find that

b¯2,3(18n+9)≡b¯2,3(6n+3)(mod8).

By mathematical induction on α, we obtain the generating function as follows:

(3.49) b¯2,32⋅3α+2n+3α+2≡b¯2,3(6n+3)(mod8).

Using (3.4) in (3.49), we get (3.13).

From (3.47), we have the generating function as follows:

(3.50) ∑n=0∞b¯2,3(18n+3)qn≡4(q;q)∞2(q2;q2)∞(q3;q3)∞6(q6;q6)∞3(mod8).

Invoking (2.1) in (3.50), we find that

(3.51) ∑n=0∞b¯2,3(18n+3)qn≡4(q2;q2)∞2(mod8).

Congruence (3.5) follows from (3.51).

Extracting the terms involving q³ⁿ from (3.43), replacing q³ by q, we have the generating function as follows:

(3.52) ∑n=0∞b¯2,3(6n+1)qn≡4(q;q)∞2(q2;q2)∞(q3;q3)∞4(q6;q6)∞2(mod8).

Invoking (2.1) in (3.52), we obtain the generating function as follows:

(3.53) ∑n=0∞b¯2,3(6n+1)qn≡4(q2;q2)∞2(mod8).

Congruence (3.3) easily follows from (3.53).

From (3.51) and (3.53), we have the generating function as follows:

(3.54) ∑n=0∞b¯2,3(36n+3)qn≡4(q;q)∞2(mod8)

and

(3.55) ∑n=0∞b¯2,3(12n+1)qn≡4(q;q)∞2(mod8).

Using the congruences (3.54) and (3.55), we get internal congruence (3.15).