Lecture hall theorems, q-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 (λ1, .... , λk) such that λ1/n ≥ λ2/n-1 ≥ ... ≥ λk/n - k + 1 ≥ 0 and we show that their generating function is Σm=0k [n m]q q(mċ1 2) (-qn-m+1;q)m/(q2n-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 Euler's theorem. The truncated anti-lectutre hall compositions are sequences (λ1,..., λk) such that λ1/n-k+1 ≥ λ2/n-k+2 ≥...≥ λk n ≥ 0. We show that their generating function is [n k]q(-qn-k+1; q)k/(q2(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.