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On the degrees of irreducible factors of higher order Bernoulli polynomials. (English) Zbl 0771.11013
For $$n\geq 0$$, the Bernoulli polynomials $$B_ n^{(\omega)}(x)$$ of order $$\omega$$ are defined by the generating function $$(t/(e^ t-1))^ \omega e^{xt}$$. Here $$\omega$$ is restricted to the range $$1,\dots,n$$. These polynomials are monic of degree $$n$$ and belong to $$\mathbb{Q}[x]$$. 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 $$2,\dots,p$$, then $$B_ n^{(\omega)}(x)$$ 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
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