Some identities and congruences concerning Euler numbers and polynomials (Q984847)

Let \(p\) be a prime, and let \(q\) be a nonnegative integer. The generalized Euler numbers \(E_n^{(q)}\) of order \(q\) are defined by the generating function \[ \left(\frac{1}{\cosh t}\right)^q=\sum_{n\geq 0}E_n^{(q)}\frac{t^n}{n!}, \] and the generalized Euler polynomials \(E_n^{(q)}(t)\) of order \(q\) are defined by \[ \left(\frac{2}{e^z+1}\right)^qe^{tz}=\sum_{n\geq 0}E_n^{(q)}(t)\frac{t^n}{n!}. \] Furthermore, let \((d_n)_{n\geq 0}\) be the sequence defined by the generating function \[ \frac{1}{2\cosh z-1}=\sum_{n\geq 0}d_n\frac{z^n}{n!}. \] This paper establishes some identities and congruences concerning \(E_n^{(q)}\), \(E_n^{(q)}(t)\) and \(d_n\), by using the properties of the moments of \(p\)-adic measures.