Cauchy-type identities through collocation matrices

TL;DR

This paper introduces a generalized framework for Cauchy identities using collocation matrices, expanding the classical identity into an infinite family with diverse applications and examples.

Contribution

The authors develop a broad, determinant-based framework that extends the classical Cauchy identity into an infinite family of identities, demonstrating its versatility through various examples.

Findings

Derived an infinite family of Cauchy-type identities

Connected classical identities to new determinant-based formulations

Showcased applications through multiple illustrative examples

Abstract

We present a broader framework for the Cauchy identity derived from the determinant expansion of collocation matrices. This approach yields an infinite family of identities, where the original Cauchy identity stands as a particular case. To illustrate the versatility and depth of this approach, we provide a range of compelling examples, showcasing the connections and applications of these novel identities.