  For an integer $k$, define poly-Euler numbers of the second kind $\widehat
E_n^{(k)}$ ($n=0,1,\dots$) by $$ \frac{{\rm Li}_k(1-e^{-4 t})}{4\sinh
t}=\sum_{n=0}^\infty\widehat E_n^{(k)}\frac{t^n}{n!}\,. $$ 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 poly-Euler numbers of the second kind, including duality
formulae and congruence relations.
