Lecture hall theorems, q-series and truncated objects

摘要

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 (lambda(1),..., lambda(k)) such thatlambda(1) greater than or equal to lambda(2)-1 greater than or equal to (...) greater than or equal to lambda(k)-k+1 greater than or equal to 0and we show that their generating function isSigma(m=0)(k) [n/m](q)q((m+1/2))(-q(n-m+1); q)(m)/(q(2n-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-lecture hall compositions are sequences (lambda(1),..., lambda(k)) such thatlambda(1)-k+1 greater than or equal to lambda(2)-k+2 greater than or equal to (...) greater than or equal to lambda(k) greater than or equal to 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. (C) 2004 Elsevier Inc. All rights reserved.