Ramanujan Type Congruences for a Partition Function

Abstract

We investigate the arithmetic properties of a certain function $b(n)$ given by $\sum\limits_{n=0}^\infty b(n)q^n=(q;q)_\infty^{-2}(q^2;q^2)_\infty^{-2}$. One of our main results is $b(9n+7)\equiv 0\ ({\rm mod\ }9)$.

Arithmetic properties of cubic and overcubic partition pairs

“…Inspired by Chan's work, Zhao and Zhong [19] studied the cubic partition pair function b(n) which is defined by…”

Arithmetic properties for cubic partition pairs modulo powers of 3

“…After Chan's work, many authors also investigated analogous partition functions. For instance, in 2011, Zhao and Zhong [8] studied the cubic partition pair function b(n) given by…”

Multiranks for Partitions into Multi-Colors

“…F. G. Garvan [19] found two other analogs the Dyson-birank and the 5-corebirank. H. Zhao and Z. Zhong [31] have also investigated the arithmetic properties of a certain function b(n) given by ∞ n=0 b(n)q n = (q; q) −2 ∞ (q 2 ; q 2 ) −2 ∞ . They have found b(5n + 4) ≡ 0 (mod 5) and b(7n + 2) ≡ b(7n + 3) ≡ b(7n + 4) ≡ b(7n + 6) ≡ 0 (mod 7) for any n 0.…”