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Euler Functions Eα(z) with Complex α and Applications

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Abstract

The aim of this paper is to extend the Euler polynomials E n (x), which may be defined in terms of their exponential generating function via

$$ \frac{{2{e^{xw}}}}{{{e^w} + 1}} = \sum\limits_{n = 0}^\infty {\frac{{{E_n}\left( x \right)}}{{n!}}} {w^n}{\text{ }}\left( {x \in \mathbb{R};\left| w \right| < \pi } \right), $$
(1)

to Euler functions E α(z) with complex indices α ∈ C; z will also be allowed to belong to C (see e.g. [26]).

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Authors and Affiliations

  1. Lehrstuhl A für Mathematik, RWTH Aachen, Templergraben 55, 52056, Aachen, Germany

    Paul L. Butzer, Stefan Flocke & Michael Hauss

Authors
  1. Paul L. Butzer
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  2. Stefan Flocke
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  3. Michael Hauss
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Editor information

Editors and Affiliations

  1. Dept. of Mathematics, Memphis State University, Memphis, Tennessee, 38152, USA

    George Anastassiou

  2. Statistics & Appl. Probability, University of California, Santa Barbara, California, 93106-3110, USA

    Svetlozar T. Rachev

Additional information

Dedicated to Jacob Korevaar on the occasion of his 70th birthday, in friendship and esteem.

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© 1994 Springer Science+Business Media New York

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Butzer, P.L., Flocke, S., Hauss, M. (1994). Euler Functions Eα(z) with Complex α and Applications. In: Anastassiou, G., Rachev, S.T. (eds) Approximation, Probability, and Related Fields. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-2494-6_9

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  • DOI: https://doi.org/10.1007/978-1-4615-2494-6_9

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4613-6063-6

  • Online ISBN: 978-1-4615-2494-6

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