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On certain generating functions in positive characteristic

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Abstract

We present new methods for the study of a class of generating functions introduced by the second author which carry some formal similarities with the Hurwitz zeta function. We prove functional identities which establish an explicit connection with certain deformations of the Carlitz logarithm introduced by M. Papanikolas and involve the Anderson-Thakur function and the Carlitz exponential function. They collect certain functional identities in families for a new class of L-functions introduced by the first author. This paper also deals with specializations at roots of unity of these generating functions, producing a link with Gauss-Thakur sums.

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Notes

  1. Where \(q^\mathbb Q=\{x\in \mathbb R_{>0};x^n=q^m,\text { for some }m,n\in \mathbb Z{\setminus }\{0\}\}\).

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Acknowledgments

Both authors wish to thank D. Goss for his continued interest in our project and helpful comments on previous versions of this work. We especially thank the anonymous referees for their careful reading and numerous helpful comments. It was their keen eyes which led us to discover a more satisfactory and correct proof of Theorem 2.

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Authors and Affiliations

  1. Institut Camille Jordan, UMR 5208, Site de Saint-Etienne, 23 rue du Dr. P. Michelon, 42023, Saint-Etienne, France

    F. Pellarin

  2. IWR, University of Heidelberg, Im Neuenheimer Feld 368, 69120, Heidelberg, Germany

    R. B. Perkins

Authors
  1. F. Pellarin
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  2. R. B. Perkins
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Corresponding author

Correspondence to R. B. Perkins.

Additional information

Communicated by A. Constantin.

Part of this research occurred while F. Pellarin was supported by the ANR HAMOT. R. B. Perkins is supported by the Alexander von Humboldt Foundation.

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Pellarin, F., Perkins, R.B. On certain generating functions in positive characteristic. Monatsh Math 180, 123–144 (2016). https://doi.org/10.1007/s00605-016-0880-6

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  • DOI: https://doi.org/10.1007/s00605-016-0880-6

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