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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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