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The classical umbral calculus and the flow of a Drinfeld module


Author: Nguyen Ngoc Dong Quan
Journal: Trans. Amer. Math. Soc. 369 (2017), 1265-1289
MSC (2010): Primary 05A40, 11G09
DOI: https://doi.org/10.1090/tran/6763
Published electronically: September 27, 2016
MathSciNet review: 3572273
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Abstract: David Goss developed a very general Fourier transform in additive harmonic analysis in the function field setting. In order to introduce the Fourier transform for continuous characteristic $ p$-valued functions on $ \mathbb{Z}_p$, Goss introduced and studied an analogue of flows in finite characteristic. In this paper, we use another approach to study flows in finite characteristic. We recast the notion of a flow in the language of the classical umbral calculus, which allows us to generalize the formula for flows first proved by Goss to a more general setting. We study duality between flows using the classical umbral calculus, and show that the duality notion introduced by Goss seems to be a natural one. We also formulate a question of Goss about the exact relationship between two flows of a Drinfeld module in the language of the classical umbral calculus, and give a partial answer to it.


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Additional Information

Nguyen Ngoc Dong Quan
Affiliation: Department of Mathematics, The University of Texas at Austin, Austin, Texas 78712
Address at time of publication: Department of Applied and Computational Mathematics and Statistics, University of Notre Dame, Notre Dame, Indiana 46556
Email: dongquan.ngoc.nguyen@gmail.com, dongquan.ngoc.nguyen@nd.edu

DOI: https://doi.org/10.1090/tran/6763
Received by editor(s): July 3, 2014
Received by editor(s) in revised form: May 25, 2015
Published electronically: September 27, 2016
Article copyright: © Copyright 2016 American Mathematical Society