Arithmetic properties of $\ell$-regular overpartition pairs

Authors : Megadahalli Sidda Mahadeva NAIKA, Chandrappa SHIVASHANKAR

Pages : 756-774

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Publication Date : 0000-00-00

Article Type : Research Paper

Abstract :In this paper, we investigate the arithmetic properties of $\ell$-regular overpartition pairs. Let $\overline{B}_{\ell}insert ignore into journalissuearticles values(n);$ denote the number of $\ell$-regular overpartition pairs of $n$. We will prove the number of Ramanujan-like congruences and infinite families of congruences modulo 3, 8, 16, 36, 48, 96 for $\overline{B}_3insert ignore into journalissuearticles values(n);$ and modulo 3, 16, 64, 96 for $\overline{B}_4insert ignore into journalissuearticles values(n);$. For example, we find that for all nonnegative integers $\alpha$ and $n$, $\overline{B}_{3}insert ignore into journalissuearticles values(3^{\alpha}insert ignore into journalissuearticles values(3n+2););\equiv 0\pmod{3}$, $\overline{B}_{3}insert ignore into journalissuearticles values(3^{\alpha}insert ignore into journalissuearticles values(6n+4););\equiv 0\pmod{3}$, and $\overline{B}_{4}insert ignore into journalissuearticles values(8n+7);\equiv 0\pmod{64}$.
Keywords : Congruences, theta function, overpartition pair, regular partition