On certain identities involving Nahm-type sums with double poles

We prove certain Nahm-type sum representations for the (odd modulus) Andrews-Gordon identities, the (even modulus) Andrews-Bressoud identities, and Rogers’ false theta functions. These identities are motivated on one hand by a recent work of C. Jennings-Shaffer and one of us on double pole series, and, on the other hand, by C'ordova, Gaiotto and Shao’s work on defect Schur’s indices.