Rectangular Scott-type Permanents

Abstract

Let $x_1,x_2,...,x_n$ be the zeroes of a polynomial P(x) of degree n and $y_1,y_2,...,y_m$ be the zeroes of another polynomial Q(y) of degree m. Our object of study is the permanent $\per(1/(x_i-y_j))_{1\le i\le n, 1\le j\le m}$, here named "Scott-type" permanent, the case of $P(x)=x^n-1$ and $Q(y)=y^n+1$ having been considered by R. F. Scott. We present an efficient approach to determining explicit evaluations of Scott-type permanents, based on generalizations of classical theorems by Cauchy and Borchardt, and of a recent theorem by Lascoux. This continues and extends the work initiated by the first author ("G\'en\'eralisation de l'identit\'e de Scott sur les permanents," to appear in Linear Algebra Appl.). Our approach enables us to provide numerous closed form evaluations of Scott-type permanents for special choices of the polynomials P(x) and Q(y), including generalizations of all the results from the above mentioned paper and of Scott's permanent itself. For example, we prove that if $P(x)=x^n-1$ and $Q(y)=y^{2n}+y^n+1$ then the corresponding Scott-type permanent is equal to $(-1)^{n+1}n!$.