On polyEuler numbers of the second kind
Abstract
For an integer $k$, define polyEuler numbers of the second kind $\widehat E_n^{(k)}$ ($n=0,1,\dots$) by $$ \frac{{\rm Li}_k(1e^{4 t})}{4\sinh t}=\sum_{n=0}^\infty\widehat E_n^{(k)}\frac{t^n}{n!}\,. $$ When $k=1$, $\widehat E_n=\widehat E_n^{(1)}$ are {\it Euler numbers of the second kind} or {\it complimentary Euler numbers} defined by $$ \frac{t}{\sinh t}=\sum_{n=0}^\infty\widehat E_n\frac{t^n}{n!}\,. $$ Euler numbers of the second kind were introduced as special cases of hypergeometric Euler numbers of the second kind in \cite{KZ}, so that they would supplement hypergeometric Euler numbers. In this paper, we give several properties of Euler numbers of the second kind. In particular, we determine their denominators. We also show several properties of polyEuler numbers of the second kind, including duality formulae and congruence relations.
 Publication:

arXiv eprints
 Pub Date:
 June 2018
 arXiv:
 arXiv:1806.05515
 Bibcode:
 2018arXiv180605515K
 Keywords:

 Mathematics  Number Theory;
 11B68;
 05A15;
 11M41
 EPrint:
 This manuscript has been accepted for publication in Bessatsu of Algebraic Number Theory and Related Topics 2016