Values of Euler polynomials (Q2741208)

The author studies values of Euler polynomials \(E_n(t)\) at rational numbers \(t=r/s\). He proves that \(s^nE_n(r/s)\in\mathbb{Z}\) if \(s\) is even and \(s^n(E_n(r/s)+(-1)^{r-1}E_n(0))\in\mathbb{Z}\) if \(s\) is odd. This generalizes slightly a result proved in a different way by \textit{G. J. Fox} [C. R. Math. Acad. Sci., Soc. R. Can. 21, 87-90 (2001; Zbl 0939.11013)]. The author also describes an interesting relation, discovered by \textit{V. Arnold} [Russ. Math. Surv. 47, No. 1, 1-51 (1992; Zbl 0791.05001)], linking the Euler and Bernoulli numbers to the so-called up-down numbers.