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Open Access
Article
Publication date: 14 December 2021

S. Shivaprasada Nayaka

Let b…

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(23α+3n+43α+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=0a(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 -regular cubic partition pairs.

Details

Arab Journal of Mathematical Sciences, vol. 29 no. 1
Type: Research Article
ISSN: 1319-5166

Keywords

Open Access
Article
Publication date: 8 March 2022

Riyajur Rahman and Nipen Saikia

Let p[1,r;t] be defined by

Abstract

Purpose

Let p[1,r;t] be defined by n=0p[1,r;t](n)qn=(E1Er)t, where t is a non-zero rational number, r ≥ 1 is an integer and Er=n=0(1qr(n+1)) for |q| < 1. The function p[1,r;t](n) is the generalisation of the two-colour partition function p[1,r;−1](n). In this paper, the authors prove some new congruences modulo odd prime by taking r = 5, 7, 11 and 13, and non-integral rational values of t.

Design/methodology/approach

Using q-series expansion/identities, the authors established general congruence modulo prime number for two-colour partition function.

Findings

In the paper, the authors study congruence properties of two-colour partition function for fractional values. The authors also give some particular cases as examples.

Originality/value

The partition functions for fractional value is studied in 2019 by Chan and Wang for Ramanujan's general partition function and then extended by Xia and Zhu in 2020. In 2021, Baruah and Das also proved some congruences related to fractional partition functions previously investigated by Chan and Wang. In this sequel, some congruences are proved for two-colour partitions in this paper. The results presented in the paper are original.

Details

Arab Journal of Mathematical Sciences, vol. 29 no. 2
Type: Research Article
ISSN: 1319-5166

Keywords

Open Access
Article
Publication date: 1 July 2021

S. Shivaprasada Nayaka, T.K. Sreelakshmi and Santosh Kumar

In this paper, the author defines the function B…

1223

Abstract

Purpose

In this paper, the author defines the function B¯i,jδ,k(n), the number of singular overpartition pairs of n without multiples of k in which no part is divisible by δ and only parts congruent to ± i, ± j modulo δ may be overlined.

Design/methodology/approach

Andrews introduced to combinatorial objects, which he called singular overpartitions and proved that these singular overpartitions depend on two parameters δ and i can be enumerated by the function C¯δ,i(n), which gives the number of overpartitions of n in which no part divisible by δ and parts ≡ ± i(Mod δ) may be overlined.

Findings

Using classical spirit of q-series techniques, the author obtains congruences modulo 4 for B¯2,48,3(n), B¯2,48,5 and B¯2,412,3.

Originality/value

The results established in this work are extension to those proved in Andrews’ singular overpatition pairs of n.

Details

Arab Journal of Mathematical Sciences, vol. 28 no. 2
Type: Research Article
ISSN: 1319-5166

Keywords

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