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arXiv · 1201.0030

On Pellarin's $L$-series

Rudolph Bronson Perkins
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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.

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Rudolph Bronson Perkins. 2011-12-29. On Pellarin's $L$-series. https://doi.org/10.1090/s0002-9939-2014-12080-6

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