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arXiv · 1207.1753

Explicit Formulae for $L$-values in Positive Characteristic

Rudolph Bronson Perkins
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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.

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Rudolph Bronson Perkins. 2014-04-30. Explicit Formulae for $L$-values in Positive Characteristic. https://doi.org/10.1007/s00209-014-1315-5

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