On Pellarin’s 𝐿-series

Perkins, Rudolph

doi:10.1090/S0002-9939-2014-12080-6

On Pellarin’s $L$-series
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by Rudolph Bronson Perkins PDF

Proc. Amer. Math. Soc. 142 (2014), 3355-3368 Request permission

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’s theory we give a 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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