# Hyperderivative power sums, Vandermonde matrices, and Carlitz multiplication coefficients

## TL;DR

This paper explores the relationships between hyperderivatives, Vandermonde matrices, and Carlitz multiplication coefficients over finite fields, providing new formulas and proofs for key theorems in function field arithmetic.

## Contribution

It introduces formulas for Carlitz multiplication coefficients via hyperderivatives and symmetric polynomials, and offers new proofs of Thakur's and Anderson's theorems using these techniques.

## Key findings

- Derived formulas for Carlitz multiplication coefficients.
- Proved identities for hyperderivative power sums.
- Provided new proofs of key theorems in function field arithmetic.

## Abstract

We investigate interconnected aspects of hyperderivatives of polynomials over finite fields, q-th powers of polynomials, and specializations of Vandermonde matrices. We construct formulas for Carlitz multiplication coefficients using hyperderivatives and symmetric polynomials, and we prove identities for hyperderivative power sums in terms of specializations of the inverse of the Vandermonde matrix. As an application of these results we give a new proof of a theorem of Thakur on explicit formulas for Anderson's special polynomials for log-algebraicity on the Carlitz module. Furthermore, by combining results of Pellarin and Perkins with these techniques, we obtain a new proof of Anderson's theorem in the general case.