Overpartitions into distinct parts without short sequences

In the first part of this paper we introduce overpartitions into distinct parts without k-sequences. When k=1 these are the partitions into parts differing by at least two which occur in the Rogers-Ramanujan identities. For general k we compute a three-variable double sum q-hypergeometric generating function and give asymptotic estimates for the number of such overpartitions. When k=2 we obtain several more double sum generating functions as well as a combinatorial identity. In the second part of the paper, we establish arithmetic and combinatorial properties of some related q-hypergeometric double sums.