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Natural Exponential Families and Umbral Calculus

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Mathematical Essays in honor of Gian-Carlo Rota

Part of the book series: Progress in Mathematics ((PM,volume 161))

Abstract

We use the Umbral Calculus to investigate the relation between natural exponential families and Sheffer polynomials. As a corollary, we obtain a new transparent proof of Feinsilver’s theorem which says that natural exponential families have a quadratic variance function if and only if their associated Sheffer polynomials are orthogonal.

Author supported NATO CRG 930554.

Author partially supported by URA CNRS 1304, EC grant CHRX-CT93-0400, the PRC Maths-Info, and NATO CRG 930554.

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Author information

Authors and Affiliations

  1. Dept. of Mathematics and Computing Science, Eindhoven University of Technology, P.O. Box 513, 5600, Eindhoven, MB, The Netherlands

    A. di Bucchianico

  2. Laboratoire Bordelais de Recherche en Informatique, Université de Bordeaux I, 33405, Talence Cedex, France

    D. E. Loeb

Authors
  1. A. di Bucchianico
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  2. D. E. Loeb
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Editor information

Editors and Affiliations

  1. Department of Mathematics, Michigan State University, 48824, East Lansing, MI, USA

    Bruce E. Sagan

  2. Department of Mathematics, MIT, 02139, Cambridge, MA, USA

    Richard P. Stanley

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© 1998 Birkhäuser

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di Bucchianico, A., Loeb, D.E. (1998). Natural Exponential Families and Umbral Calculus. In: Sagan, B.E., Stanley, R.P. (eds) Mathematical Essays in honor of Gian-Carlo Rota. Progress in Mathematics, vol 161. Birkhäuser Boston. https://doi.org/10.1007/978-1-4612-4108-9_9

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  • DOI: https://doi.org/10.1007/978-1-4612-4108-9_9

  • Publisher Name: Birkhäuser Boston

  • Print ISBN: 978-1-4612-8656-1

  • Online ISBN: 978-1-4612-4108-9

  • eBook Packages: Springer Book Archive

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