Periodica Mathematica Hungarica

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

Complementary Euler numbers

  • Takao KomatsuEmail author


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.


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



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


  1. 1.
    T. Arakawa, M. Kaneko, Multiple zeta values, poly-Bernoulli numbers, and related zeta functions. Nagoya Math. J. 153, 189–209 (1999)CrossRefMathSciNetzbMATHGoogle Scholar
  2. 2.
    M. Kaneko, Poly-Bernoulli numbers. J. Theor. Nombres Bordx. 9, 221–228 (1997)CrossRefMathSciNetzbMATHGoogle Scholar
  3. 3.
    T. Komatsu, Poly-Cauchy numbers. Kyushu J. Math. 67, 143–153 (2013)CrossRefMathSciNetzbMATHGoogle Scholar
  4. 4.
    T. Komatsu, H. Zhu, Hypergeometric Euler numbers (2016). arXiv:1612.06210
  5. 5.
    S. Koumandos, H. Laurberg Pedersen, Turán type inequalities for the partial sums of the generating functions of Bernoulli and Euler numbers. Math. Nachr. 285, 2129–2156 (2012)CrossRefMathSciNetzbMATHGoogle Scholar
  6. 6.
    N.J.A. Sloane, The on-line encyclopedia of integer sequences. (2017)Google Scholar
  7. 7.
    Y. Ohno, Y. Sasaki, On the parity of poly-Euler numbers. RIMS Kokyuroku Bessatsu B32, 271–278 (2012)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Y. Ohno, Y. Sasaki, Periodicity on poly-Euler numbers and Vandiver type congruence for Euler numbers. RIMS Kokyuroku Bessatsu B44, 205–211 (2013)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Y. Ohno, Y. Sasaki, On poly-Euler numbers. J. Aust. Math. Soc. (2016). doi: 10.1017/S1446788716000495 zbMATHGoogle Scholar
  10. 10.
    Y. Sasaki, On generalized poly-Bernoulli numbers and related \(L\)-functions. J. Number Theory 132, 156–170 (2012)CrossRefMathSciNetzbMATHGoogle Scholar

Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2017

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsWuhan UniversityWuhanChina

Personalised recommendations