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On Pellarin's $ L$-series


Author: Rudolph Bronson Perkins
Journal: Proc. Amer. Math. Soc. 142 (2014), 3355-3368
MSC (2010): Primary 11M38
DOI: https://doi.org/10.1090/S0002-9939-2014-12080-6
Published electronically: June 16, 2014
MathSciNet review: 3238413
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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'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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Rudolph Bronson Perkins
Affiliation: Department of Mathematics, The Ohio State University, 231 West 18th Avenue, Columbus, Ohio 43210
Email: perkins@math.osu.edu

DOI: https://doi.org/10.1090/S0002-9939-2014-12080-6
Received by editor(s): December 29, 2011
Received by editor(s) in revised form: February 21, 2012, and October 18, 2012
Published electronically: June 16, 2014
Communicated by: Matthew A. Papanikolas
Article copyright: © Copyright 2014 American Mathematical Society

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