On Pellarin's $L$-series

TL;DR

This paper investigates the zeros of Pellarin's $L$-series, providing conditions for trivial zeros, analyzing their simplicity, and exploring associated polynomial degrees and Carlitz polynomial approximations.

Contribution

It introduces new criteria for trivial zeros of Pellarin's $L$-series, determines the degree of related special polynomials, and extends Carlitz polynomial theory for additive and linear functions.

Findings

Trivial zeros are characterized by necessary and sufficient conditions.

All trivial zeros identified are shown to be simple zeros.

Exact degrees of special polynomials associated with Pellarin's $L$-series are determined.

Abstract

Necessary and sufficient conditions are given for a negative integer to be a trivial zero of a new type of $L$-series recently discovered by F. Pellarin, and it is shown that any such trivial zero is simple. We determine the exact degree of the special polynomials associated to Pellarin's $L$-series. The theory of Carlitz polynomial approximations is developed further for both additive and $\mathbb{F}_q$-linear functions. Using Carlitz' theory we give generating series for the power sums occurring as the coefficients of the special polynomials associated to Pellarin's series, and a connection is made between the Wagner representation for $\chi_t$ and the value of Pellarin's $L$-series at 1.