Identities and congruences involving higher-order Euler-Bernoulli numbers and polynomials (Q2735229)

Let the Euler numbers \(E_{2n}\) be defined by the series \(\operatorname {sec}t= \sum_{n=0}^\infty E_{2n} \frac{t^{2n}} {(2n)!}\) and, generally, the \(k\)th-order Euler numbers \(E_{2n}^{(k)}\) by \((\operatorname {sec}t)^k= \sum_{n=0}^\infty E_{2n}^{(k)} \frac{t^{2n}} {(2n)!}\), where \(|t|< \frac{\pi}{2}\) and \(k\) is a positive integer, \textit{W. Zhang} [Fibonacci Q. 36, 154-157 (1998; Zbl 0919.11018)] expressed \(E_{2n}^{(2m+1)}\) as a linear combination of Euler numbers. In this paper, the author expresses \(E_{2n}^{(2m)}\) as a linear combination of second-order Euler numbers. As corollaries, some congruences on second-order Euler numbers and Bernoulli numbers are obtained.