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)(λ1,…,λn) satisfyingk≥λ1a1≥λ2a2≥…≥λnan≥0,where a:=(a1,…,an)=(1,2,…,n)a:=(a1,…,an)=(1,2,…,n) or (n,n−1,…,1)(n,n−1,…,1). When k→∞k→∞ we recover the refined anti-lecture hall and lecture hall theorems. When a=(1,2,…,n)a=(1,2,…,n) and n→∞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.