A Pfaffian-Hafnian analogue of Borchardt's identity

Masao Ishikawa; Hiroyuki Kawamuko; Soichi Okada

A Pfaffian-Hafnian analogue of Borchardt's identity

Masao Ishikawa, Hiroyuki Kawamuko, Soichi Okada

Research output: Contribution to journal › Article › peer-review

Abstract

We prove Pf (x_i - x_j/(x_i + x _j)²)_1≤i,j≤2n = π _1≤i<j≤2n x_i - _j/x_i + _j · Hf (1/x_i + x_j) _1≤i,j≤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.

Original language	English
Journal	Electronic Journal of Combinatorics
Volume	12
Issue number	1 N
Publication status	Published - Jun 14 2005
Externally published	Yes

ASJC Scopus subject areas

Theoretical Computer Science
Geometry and Topology
Discrete Mathematics and Combinatorics
Computational Theory and Mathematics
Applied Mathematics

Cite this

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abstract = "We prove Pf (xi - xj/(xi + x j)2)1≤i,j≤2n = π 1≤i xi - j/xi + j · Hf (1/xi + xj) 1≤i,j≤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.",

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N2 - We prove Pf (xi - xj/(xi + x j)2)1≤i,j≤2n = π 1≤i xi - j/xi + j · Hf (1/xi + xj) 1≤i,j≤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.

AB - We prove Pf (xi - xj/(xi + x j)2)1≤i,j≤2n = π 1≤i xi - j/xi + j · Hf (1/xi + xj) 1≤i,j≤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.

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A Pfaffian-Hafnian analogue of Borchardt's identity

Abstract

ASJC Scopus subject areas

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