Generalization of the new antisymmetrizer-to-determinant theorem

Establish a generalization of Theorem antisymtodet for the Cauchy-type antisymmetrizer identity presented in Theorem cauchyg, and determine whether such a generalization has applications analogous to those of the new antisymmetrizer-to-determinant formula.

Background

The paper proves a common generalization of the Cauchy determinant evaluation and an earlier antisymmetrizer-to-determinant lemma. The authors then ask whether the more substantial identity established in their main theorem, Theorem antisymtodet, admits an analogous extension within this generalized Cauchy framework.

The requested result would connect the new theorem to the broader rational-function identity of Theorem cauchyg and could potentially yield further applications. The paper leaves both the existence of such a generalization and its usefulness unresolved.

References

In Section~\ref{cauchysec}, we present a common generalization of Lemma~\ref{first} and the Cauchy determinant evaluation and pose it as an open problem to find a similar generalization of Theorem~\ref{antisymtodet}.

— A new antisymmetrizer-to-determinant formula and a conjecture of Colomo and Pronko  (2608.24548 - Fischer et al., 25 Aug 2026) in Section 3, “A generalization of the Cauchy determinant,” final paragraph