Periodica Mathematica Hungarica

, Volume 75, Issue 2, pp 302–314

# Complementary Euler numbers

• Takao Komatsu
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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