Values of a class of generalized Euler and Bernoulli numbers. (Q2881242)

The authors consider generalized Euler numbers \(\varepsilon _n(u)\) and generalized Bernoulli numbers \(\beta _n(u)\) defined by generating series NEWLINE\[NEWLINE \frac {2ue^x}{e^{2x}+u}=\sum _{n=0}^{\infty } \varepsilon _n(u) \frac {x^n}{n!} \qquad \text{and} \qquad \frac {uxe^{(u-1)x}}{e^{x}-u}=\sum _{n=0}^{\infty } \beta _n(u) \frac {x^n}{n!}. NEWLINE\]NEWLINE They prove that for integers \(r>s>0\) and \(n\geq 0\) the expressions \((r+s)^{n+1}\varepsilon _n(\frac {r}{s})\), \((r+s)^{n+1}(\varepsilon _n(\frac {r}{s})-E_n)\), \((r+s)^{n+1}(\varepsilon _n(\frac {r}{s})+E_n)\), where \(E_n\) is the Euler number, and \((s-r)^ns^n\beta _n(\frac {r}{s})\) are integers. They also discuss parity of these expressions and some of their divisors.