*(English)*Zbl 0771.11013

For $n\ge 0$, the Bernoulli polynomials ${B}_{n}^{\left(\omega \right)}\left(x\right)$ of order $\omega $ are defined by the generating function ${(t/({e}^{t}-1))}^{\omega}{e}^{xt}$. Here $\omega $ is restricted to the range $1,\cdots ,n$. These polynomials are monic of degree $n$ and belong to $\mathbb{Q}\left[x\right]$. The author obtains irreducibility results for them by studying the polynomials ${A}_{n}(x,s)={B}_{n}^{(n+s+1)}(x+1)$ generated by the functions ${(1+t)}^{x}{({t}^{-1}ln(1+t))}^{s}$. His method is to study, for a fixed prime $p$, the powers of $p$ dividing the denominators of the coefficients of ${A}_{n}(x,s)$; these $p$-powers are in fact characterized in terms of the $p$-adic expansion of $n$. The results generalize previous ones by N. Kimura and P. J. McCarthy.

A typical result: If $p$ is odd and $(n-1)/(p-1)$ is an integer in the interval $2,\cdots ,p$, then ${B}_{n}^{\left(\omega \right)}\left(x\right)$ has an irreducible factor of degree at least $n-p$ iff a given inequality depending on $n$, $p$, and $\omega $ holds true.

##### MSC:

11B68 | Bernoulli and Euler numbers and polynomials |