A Pfaffian–Hafnian Analogue of Borchardt's Identity

Abstract

We prove $$ {\rm Pf}\! \left( { x_i - x_j \over (x_i + x_j)^2 } \right)_{1 \le i, j \le 2n} = \prod_{1 \le i < j \le 2n}{ x_i - x_j \over x_i + x_j } {\rm Hf}\! \left( { 1 \over x_i + x_j } \right)_{1 \le i, j \le 2n} $$ (and its variants) by using complex analysis. This identity can be regarded as a Pfaffian–Hafnian analogue of Borchardt's identity and as a generalization of Schur's identity.