Explicit Formulae for $L$-values in Positive Characteristic

TL;DR

This paper derives explicit formulas for Pellarin's $L$-series values in positive characteristic, connecting them to special functions and providing recursive relations and divisibility properties.

Contribution

It introduces a closed-form formula for generating series of Pellarin's $L$-values using interpolation polynomials and special functions, advancing understanding in positive characteristic number theory.

Findings

Explicit formulas for Pellarin's $L$-series values

Recursive relations for $L$-series

Divisibility results for Bernoulli-Carlitz numbers

Abstract

We focus on the generating series for the rational special values of Pellarin's $L$-series in $1 \leq s \leq 2(q-1)$ indeterminates, and using interpolation polynomials we prove a closed form formula relating this generating series to the Carlitz exponential, the Anderson-Thakur function, and the Anderson generating functions for the Carlitz module. We draw several corollaries, including explicit formulae and recursive relations for Pellarin's $L$-series in the same range of $s$, and divisibility results on the numerators of the Bernoulli-Carlitz numbers by monic irreducibles of degrees one and two.