Abstract
Purpose
Let
Design/methodology/approach
H.C. Chan has studied the congruence properties of cubic partition function a(n), which is defined by
Findings
To establish several congruence modulo 8 for
Originality/value
The results established in the work are extension to those proved in ℓ-regular cubic partition pairs.
Keywords
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
Publisher
:Emerald Publishing Limited
Copyright © 2021, S. Shivaprasada Nayaka
License
Published in Arab Journal of Mathematical Sciences. Published by Emerald Publishing Limited. This article is published under the Creative Commons Attribution (CC BY 4.0) license. Anyone may reproduce, distribute, translate and create derivative works of this article (for both commercial and non-commercial purposes), subject to full attribution to the original publication and authors. The full terms of this license may be seen at http://creativecommons.org/licences/by/4.0/legalcode
1. Introduction
A partition λ of a natural number n is a finite non-increasing sequence of positive integer parts λi (1 ≤ i ≤ m) such that
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:
Ramanujan’s three famous congruences of p(n) are as follows:
In [1–3], H.C. Chan has studied the congruence properties of cubic partition function a(n), which is defined by as follows:
B. Kim [4] studied its overpartition analog, the overcubic partition function
In [5], M.D. Hirschhorn obtained the results satisfied by
Recently, Kim [8] studied congruence properties of
More recently, Lin [9] studied various arithmetic properties of
In [10], Naika and Nayaka have established some congruences for ℓ-regular cubic partition pairs. Let
In this paper, we establish several congruences modulo 8 for
2. Preliminaries
For |ab| < 1, Ramanujan’s general theta function f(a, b) is defined as follows:
Some special cases of f(a, b) are as follows:
Where the product representation of f(a, b) arises from Jacobi’s triple product identity [11, p. 35, Entry 19] as follows:
The following dissection formulas to prove our main results.
For each prime p and n ≥ 1,
The following 2-dissections holds:
Lemma (2.2) is a consequence of dissection formulas of Ramanujan, which is collected in Berndt’s book [11, p. 40, Entry 25].
The following 2-dissections holds:
Hirschhorn, Garvan and Borwein [12] proved (2.4) and (2.5). For proof of (2.6), see [13].
The following 2-dissections holds:
Eqn (2.7) was proved by Baruah and Ojah [14]. Replacing q by − q in (2.7) and using the fact that
The following 3-dissection hold:
One can see this identity in [15].
[11, p. 345, Entry 1 (iv)]. We have the following 3-dissection
3. Congruences modulo 8 for b ¯ 2,3 ( n )
For each α ≥ 0 and n ≥ 1, we have
Employing (2.4) and (2.5) in (1.1), we have
Invoking (2.1) in (3.18), we obtain the generating function as follows:
Substituting (2.8) into (3.19), we get the generating function as follows:
Extracting the terms in which powers of q are congruent to 1 modulo 2 from (3.20), we have the generating function as follows:
Extracting the terms involving q3n from (3.22), replacing q3 by q, we have the generating function as follows:
Employing (2.2) into (3.23), we find the generating function as follows:
Extracting the terms involving q2n from (3.24), replacing q2 by q, we have the generating function as follows:
Invoking (2.1) in (3.25), we get the generating function as follows:
Ramanujan recorded the following identity in his third note book; for proof, one can see [11, p. 49].
Substituting (3.27) into (3.26), we obtain the generating function as follows
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:
The results (3.9) and (3.10) follow from (3.29).
From (3.29), we obtain the generating function as follows:
Using the congruences (3.30) and (3.26), we can see that
By mathematical induction on α, we find that
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:
Invoking (2.1) in (3.32), we get the generating function as follows:
From (3.20), we can see that
Invoking (2.1) in (3.34), we have the generating function as follows:
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:
Employing (2.9) into (3.36), we obtain the generating function as follows:
Extracting the terms in which powers of q are congruent to 1 modulo 3 from (3.37), we have the generating function as follows:
The results (3.6) and (3.8) follow from (3.38).
From (3.38), we have the generating function as follows:
Using the congruences (3.39) and (3.36), we can see that
By mathematical induction on α, we obtain the generating function as follows:
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:
Invoking (2.1) in (3.41), we get the generating function as follows:
Using the congruences (3.33) and (3.42), we obtain (3.14).
From (3.19), it can be rewritten as follows:
Employing (2.9) into (3.43), we obtain the generating function as follows:
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:
Substituting (2.10) into (3.46), we obtain the generating function as follows:
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:
Using the congruences (3.48) and (3.46), we find that
By mathematical induction on α, we obtain the generating function as follows:
Using (3.4) in (3.49), we get (3.13).
From (3.47), we have the generating function as follows:
Invoking (2.1) in (3.50), we find that
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:
Invoking (2.1) in (3.52), we obtain the generating function as follows:
Congruence (3.3) easily follows from (3.53).
From (3.51) and (3.53), we have the generating function as follows:
Using the congruences (3.54) and (3.55), we get internal congruence (3.15).
References
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