Anti-lecture hall compositions and Andrews' generalization of the Watson-Whipple transformation

For fixed n and k, we find a three-variable generating function for the set of sequences (λ1, . . . , λn) satisfying k ≥ λ1 a1 ≥ λ2 a2 ≥ . . . ≥ λn an ≥ 0, where a := (a1, . . . , an) = (1, 2, . . . , n) or (n, n − 1, . . . , 1). When k → ∞ we recover the refined anti-lecture hall and lecture hall theorems. When a = (1, 2, . . . , n) and n → ∞, we obtain a refinement of a recent result of Chen, Sang and Shi. The main tools are elementary combinatorics and Andrews’ generalization of the Watson-Whipple transformation.