Title: Some Properties of Generalized Euler Numbers
Abstract: We define infinitely many sequences of integers one sequence for each positive integer k ≦ 2 by (1.1) where are the k -th roots of unity and ( E (k) ) n is replaced by E n (k) after multiplying out. An immediate consequence of (1.1) is (1.2) Therefore, we are interested in numbers of the form E sk (k) ( s = 0, 1, 2, …; k = 2, 3, …). Some special cases have been considered in the literature. For k = 2, we obtain the Euler numbers (see e.g. [ 8 ]). The case k = 3 is considered briefly by D. H. Lehmer [ 7 ], and the case k = 4 by Leeming [ 6 ] and Carlitz ([ 1 ] and [ 2 ]).