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Periodica Mathematica Hungarica

, Volume 75, Issue 2, pp 302–314 | Cite as

Complementary Euler numbers

  • Takao KomatsuEmail author
Article

Abstract

For an integer k, define poly-Euler numbers of the second kind \(\widehat{E}_n^{(k)}\) (\(n=0,1,\ldots \)) by
$$\begin{aligned} \frac{{\mathrm{Li}}_k(1-e^{-4 t})}{4\sinh t}=\sum _{n=0}^\infty \widehat{E}_n^{(k)}\frac{t^n}{n!}. \end{aligned}$$
When \(k=1\), \(\widehat{E}_n=\widehat{E}_n^{(1)}\) are Euler numbers of the second kind or complimentary Euler numbers defined by
$$\begin{aligned} \frac{t}{\sinh t}=\sum _{n=0}^\infty \widehat{E}_n\frac{t^n}{n!}. \end{aligned}$$
Euler numbers of the second kind were introduced as special cases of hypergeometric Euler numbers of the second kind in Komatsu and Zhu (Hypergeometric Euler numbers, 2016, arXiv:1612.06210), so that they would supplement hypergeometric Euler numbers. In this paper, we study generalized Euler numbers of the second kind and give several properties and applications.

Keywords

Euler numbers Complementary Euler numbers Euler numbers of the second kind Poly-Euler numbers of the second kind Determinant Zeta functions 

Notes

Acknowledgements

The author thanks the anonymous referee for careful reading of the manuscript and giving the hint to Theorem 4.3.

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Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2017

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsWuhan UniversityWuhanChina

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