A combinatorial determinant

TL;DR

This paper explores a theorem by Mina that evaluates determinants of matrices with derivatives of functions, providing simplified proofs for special and general cases, and discussing applications.

Contribution

It offers simplified proofs for Mina's determinant evaluation theorem and extends the results to the general case, highlighting key applications.

Findings

Simplified proof of Mina's determinant theorem at x=0

Extension to the general case of the theorem

Identification of applications of the determinant evaluation

Abstract

A theorem of Mina evaluates the determinant of a matrix with entries $D^j(f(x)^i)$. We note the important special case where the matrix entries are evaluated at $x=0$ and give a simple proof of it, and some applications. We then give a short proof of the general case.