An elliptic generalization of Schur's Pfaffian identity

TL;DR

This paper introduces an elliptic function-based Pfaffian identity that generalizes Schur's classical Pfaffian identity, extending the mathematical framework to elliptic functions and connecting it to Frobenius and Cauchy's identities.

Contribution

It presents a novel elliptic Pfaffian identity that generalizes Schur's Pfaffian identity and relates to elliptic versions of classical determinant identities.

Findings

Derived a new elliptic Pfaffian identity

Connected the identity to elliptic generalizations of Frobenius and Cauchy's identities

Provided a rational limit that recovers known identities

Abstract

We present a Pfaffian identity involving elliptic functions, whose rational limit gives a generalization of Schur's Pfaffian identity for Pf ((x_j - x_i)/(x_j + x_i)). This identity is regarded as a Pfaffian counterpart of Frobenius identity, which is an elliptic generalization of Cauchy's determinant identity for det (1/ (x_i + x_j)).