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On the degrees of irreducible factors of higher order Bernoulli polynomials. (English) Zbl 0771.11013

For n0, the Bernoulli polynomials B n (ω) (x) of order ω are defined by the generating function (t/(e t -1)) ω e xt . Here ω is restricted to the range 1,,n. These polynomials are monic of degree n and belong to [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,,p, then B n (ω) (x) has an irreducible factor of degree at least n-p iff a given inequality depending on n, p, and ω holds true.


MSC:
11B68Bernoulli and Euler numbers and polynomials