Congruences For Sequences Analogous To Euler Numbers

For a given real number a we define the sequence {En,a} by E0,a= 1 and En,a=-a [n/2]k=1 n2 k En-2k,a(n ≥ 1), where [x] is the greatest integer not exceeding x. Since En,1= Enis the n-th Euler number, En,acan be viewed as a natural generalization of Euler numbers. In this paper we deduce some identities and an inversion formula involving {En,a} and {En,a(x)}, where En,a(x) = n k=0 n k Ek,axn-k, and establish congruences for E2 n,a(mod 2ord2n+8),E2mk+b,a-Eb,a(mod 2m), E2 n,a(mod 3ord3n+5)and E2 n,a(mod 5ord5n+4) provided that a is a nonzero integer, k and m are positive integers and b ∈ {0, 2, 4,...}, where ordpn is the least nonnegative integer α such that pα| n but pα+1 n.