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On the degrees of irreducible factors of higher order Bernoulli polynomials. (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 ${\left(t/\left({e}^{t}-1\right)\right)}^{\omega }{e}^{xt}$. Here $\omega$ is restricted to the range $1,\cdots ,n$. These polynomials are monic of degree $n$ and belong to $ℚ\left[x\right]$. The author obtains irreducibility results for them by studying the polynomials ${A}_{n}\left(x,s\right)={B}_{n}^{\left(n+s+1\right)}\left(x+1\right)$ generated by the functions ${\left(1+t\right)}^{x}{\left({t}^{-1}ln\left(1+t\right)\right)}^{s}$. His method is to study, for a fixed prime $p$, the powers of $p$ dividing the denominators of the coefficients of ${A}_{n}\left(x,s\right)$; 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 $\left(n-1\right)/\left(p-1\right)$ 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