Congruences for Overpartition Pairs with Restricted Odd Differences

Let b^-(k) (n) denote the number of overpartition pairs of n where (i) consecutive parts di?er by a multiple of k + 1 unless the larger of the two is overlined, and (ii) the smallest part is overlined unless it is divisible by k+1. We prove many in?nite families of congruences modulo powers of 2 and 3 for b^-(2) (n) and congruences modulo 4 and 5 for b^-(4) (n). For example, for all n ? 0 and ?,? ? 0,

b^-(4)(4·3^4? ·5^2?(5n + i) + 3^4? ·5^2?)? 0 (mod 5),

where i = 3,4.