Some Properties of Partitions in Terms of Crank

Let N(r,m, n) (resp. M(r,m, n)) denote the number of partitions of n whose ranks (resp. cranks) are congruent to r modulo m. Atkin and Swinnerton-Dyer gave the relations between the numbers N(r,m,mn+k) when m = 5, 7 and 0 ≤ r, k < m. Garvan gave the relations between the numbers M(r,m,mn+k) when m = 5, 7, and 11, 0 ≤ r, k < m. Here, we show that the methods of Atkin and Swinnerton-Dyer can be extended to prove the relations for the crank.