Lecture hall theorems, -series and truncated objects

We show here that the refined theorems for both lecture hall partitions and anti-lecture hall compositions can be obtained as straightforward consequences of two q -Chu Vandermonde identities, once an appropriate recurrence is derived. We use this approach to get new lecture hall-type theorems for truncated objects. The truncated lecture hall partitions are sequences ( λ 1 , … , λ k ) such that λ 1 n ⩾ λ 2 n - 1 ⩾ ⋯ ⩾ λ k n - k + 1 ⩾ 0 and we show that their generating function is ∑ m = 0 k n m q q m + 1 2 ( - q n - m + 1 ; q ) m ( q 2 n - m + 1 ; q ) m . From this, we are able to give a combinatorial characterization of truncated lecture hall partitions and new finite versions of refinements of Eulerʹs theorem. The truncated anti-lecture hall compositions are sequences ( λ 1 , … , λ k ) such that λ 1 n - k + 1 ⩾ λ 2 n - k + 2 ⩾ ⋯ ⩾ λ k n ⩾ 0 . We show that their generating function is n k q ( - q n - k + 1 ; q ) k ( q 2 ( n - k + 1 ) ; q ) k , giving a finite version of a well-known partition identity. We give two different multivariate refinements of these new results: the q -calculus approach gives ( u , v , q ) -refinements, while a completely different approach gives odd / even ( x , y ) -refinements.