The ranks of (a,b)-Fibonacci sequences and congruences for certain partition functions and Ramanujan's mock theta functions

In this paper, employing some identities due to Newman, we present a new method for discovering infinite families of congruences and strange congruences for c(n) which is defined by Sigma(n = 0) (infinity) c (n) q(n) = Pi(k = 1) (infinity) (1-q(k))r(1-q(kt))(s) . Here r,s,t are some integers with t>1 . By our method, in order to discover infinite families of congruences modulo M for c(n), it suffices to compute the ranks of (a,b) -Fibonacci sequences modulo M. As applications, we establish many infinite families of congruences and strange congruences for some Ramanujan's mock theta functions and certain partition functions, such as, a partition function related to mock theta function omega (q) and broken k-diamond partitions.