Congruences related to Eulerian and Euler numbers

Abstract

Let H k denote the harmonic number of order k and H ′ p−1 denote the sum of order p − 1 of the odd reciprocals of integers. Given an odd prime number p > 3, by using in particular some congruences on Eulerian numbers, we show the following congruences:Let T k denote the k-th tangent number defined by the series expansionGiven an old prime number p > 3, by using in particular some congruence on the Euler numbers, we show the congruence:p−1 2 k=1 (−1) k (2k − 1)T 2k−1 = 0 mod p if p ≡ 1 mod 4 2 mod p if p ≡ 3 mod 4