An extension of the Abel-Liouville identity

TL;DR

This paper extends the Abel-Liouville identity using noncommutative Bell polynomials to generalize Wronskians and characterizes when vector-valued functions are equivalent based on their Wronskians.

Contribution

It introduces a novel extension of the Abel-Liouville identity employing noncommutative Bell polynomials and provides a characterization of range equivalence for certain vector functions.

Findings

Extended Abel-Liouville identity with noncommutative Bell polynomials

Characterization of range equivalence for n-dimensional functions

Insights into generalized Wronskians

Abstract

In this note, we present an extension of the celebrated Abel-Liouville identity in terms of noncommutative complete Bell polynomials for generalized Wronskians. We also characterize the range equivalence of $n$-dimensional vector-valued functions in the subclass of $n$-times differentiable functions with a nonvanishing Wronskian.