New permanent approximation inequalities via identities

The aim of this paper is to present new upper bounds for the distance between a properly normalized permanent of a rectangular complex matrix and the product of the arithmetic means of the entries of its columns. It turns out that the bounds improve on those from earlier work. Our proofs are based on some new identities for the above-mentioned differences and also for related expressions for matrices over a rational associative commutative unital algebra. Some of our identities are generalizations of results in Dougall (Proceedings of the Edinburgh Mathematical Society, 24, 61-77, 1905). Second order results are also included.