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

S Shivaprasada Nayaka

doi:10.1108/ajms-07-2021-0162

Abstract

PurposeLet 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/approachH.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)∞.FindingsTo 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/valueThe results established in the work are extension to those proved in ℓ-regular cubic partition pairs.

Highlights

A partition λ of a natural number n is a finite non-increasing sequence of positive integer parts λi (1 ≤ i ≤ m) such that n 1⁄4 λ1 þ λ2 þ λ3 þÁÁÁþ λm: In this case, we write jλj 5 n
Using the congruences (3.30) and (3.26), we can see that b2;3ð216n þ 27Þ ≡ b2;3ð24n þ 3Þ ðmod 8Þ: By mathematical induction on α, we find that b2;3ð216$9αn þ 27$9αÞ ≡ b2;3ð24n þ 3Þ ðmod 8Þ: (3.31)
Extracting the terms in which powers of q are congruent to 1 modulo 3 from (3.37), we have the generating function as follows: X ∞ b2;3ð24n þ 9Þqn ≡ 4ðq3; q3Þ∞ðq6; q6Þ∞ ðmod 8Þ: n1⁄40 (3.38)

Summary

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 n 1⁄4 λ1 þ λ2 þ λ3 þÁÁÁþ λm: In this case, we write jλj 5 n. Ðq; 1 qÞ2∞ðq2; q2Þ2∞: Recently, Kim [8] studied congruence properties of bðnÞ, which denotes overcubic partition pairs of n, whose generating function is given by as follows: X ∞ bðnÞqn n1⁄40

Objectives

Results

Conclusion