A Crank Analog on a Certain Kind of Partition Function Arising from the Cubic Continued Fraction

In a series of papers, H.-C. Chan has studied congruence properties of a certain kind of partition function that arises from Ramanujan’s cubic continued fraction. This partition function a(n), is defined by ∑∞ n=0 a(n)q n = 1 (q;q)∞(q2;q2)∞ . In particular, he proved that a(3n + 2) ≡ 0 (mod 3). As Chan mentioned in his paper, it is natural to ask if there exists an analog of the rank or the crank for the ordinary partition function that provides a combinatorial explanation of the above congruence. Here, we will define a crank analog M ′(m,N, n) for a(n) and prove that M ′(0, 3, 3n + 2) ≡ M ′(1, 3, 3n + 2) ≡ M ′(2, 3, 3n + 2) (mod 3), for all nonnegative integers n, where M ′(m,N, n) is the number of partitions of n with crank ≡ m (mod N). Next, using the theory of modular forms, we will investigate further congruences of a(n).