Euler tangent numbers modulo 720 and Genocchi numbers modulo 45

Abstract

: We establish congruences for higher order Euler polynomials modulo 720. We apply this result for constructing analogues of Stern congruences for Euler secant numbers E 4 n (cid:3) 5 ð mod 60 Þ ; E 4 n þ 2 (cid:3) (cid:4) 1 ð mod 60 Þ to Euler tangent numbers and Genocchi numbers. We prove that Euler tangent numbers satisfy the following congruences E 4 n þ 1 (cid:3) 16 ð mod 720 Þ , and E 4 n þ 3 (cid:3) (cid:4) 272 ð mod 720 Þ . We establish 12-periodic property of Genocchi numbers modulo 45.