Ramanujan's Cubic Continued Fraction and Ramanujan Type Congruences for a Certain Partition Function

Abstract

In this paper, we study the divisibility of the function a(n) defined by [Formula: see text]. In particular, we prove certain "Ramanujan type congruences" for a(n) modulo powers of 3.

From a Ramanujan-Selberg continued fraction to a Jacobian identity

“…Recently, the following was discovered. Denote the Ramanujan cubic continued fraction by := [12]; see also its implications in [13,14]). Second, we remark that the strategy used in this paper can also be applied to the Ramanujan-Göllnitz-Gordon continued fraction (see p. 229 of Ramanujan's second notebook [25]; see also [1,20,21]; and also the paper by Chan and Huang [17], in which the authors provided a comprehensive theory of the Ramanujan-Göllnitz-Gordon continued fraction) and proved certain -identities.…”

A new proof of two identities involving Ramanujan’s cubic continued fraction

“…Now, we combine the two ways of writing K ∞ (a) to prove our lemma. Indeed, (15) and (18) = 1 3(q; q) ∞ J (1) + J (β) + J β 2 by (19).…”

“…A consequence of Theorem 2 is Theorem 3 (Chan, Theorem 1 in [19]) For k ≥ 1 and n being nonnegative,…”

New congruences for overcubic partition pairs

“…C. Chan in his papers [3,4,5] studied the congruence properties of the cubic partition function a(n), which is defined by ∞ n=0 a(n)q n = 1 (q; q) ∞ (q 2 ; q 2 ) ∞ = 1 f 1 f 2 .…”