On poly-Euler numbers of the second kind (Algebraic Number Theory and Related Topics 2016)

Abstract

For an integer k, define poly-Euler numbers of the second kind Ên(k) (n = 0, 1, ... ) by Lik(1-e -4t)/4 sinh t = ∞Σ n=0 Ên(k) tn/n!. When k = 1, Ên = Ên(1) are Euler numbers of the second kind or complimentary Euler numbers defined by t/sinh t = ∞Σ n=0 Ên tn/n!. Euler numbers of the second kind were introduced as special cases of hypergeometric Euler numbers of the second kind in [7], 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.