Geodesic
A Pfaffian-Hafnian analogue of Borchardt's identity
The electronic journal of combinatorics, Tome 12 (2005)

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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.
DOI : 10.37236/1976
Classification : 05E05
Mots-clés : Schur's identity 
Masao Ishikawa; Hiroyuki Kawamuko; Soichi Okada. A Pfaffian-Hafnian analogue of Borchardt's identity. The electronic journal of combinatorics, Tome 12 (2005). doi: 10.37236/1976
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     author = {Masao Ishikawa and Hiroyuki Kawamuko and Soichi Okada},
     title = {A {Pfaffian-Hafnian} analogue of {Borchardt's} identity},
     journal = {The electronic journal of combinatorics},
     year = {2005},
     volume = {12},
     doi = {10.37236/1976},
     zbl = {1074.05090},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/1976/}
}
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