Partitions with bounded differences between largest and smallest parts

In [3] Breuer and Kronholm gave in effect two proofs for an explicit formula for the generating function for partitions where the difference between largest and smallest part is bounded by a given integer t. Their first proof is geometric, involving counting lattice points within a polyhedral region; their second proof constructs an explicit bijection. In this paper we give another proof, a formal calculation involving elementary q-series manipulation, involving no results deeper than the q-binomial theorem. The results of [3] imply a theorem of Andrews, Beck and Robbins [2] on partitions where the difference between largest and smallest part is a fixed integer t. They use formal q-series methods which go beyond ours, for instance Heine’s transformation for basic hypergeometric series.