Summary: On the degrees of irreducible factors of higher order Bernoulli polynomials
ARNOLD ADELBERG (Grinnell College)
1. Introduction. In this paper, we generalize the current results on the p--Eisenstein
behavior of first and higher order Bernoulli polynomials [4, 6, 7, 8, 9], using the machinery
of . In so doing, we provide a broader framework for the known results, all of which
are either immediate consequences or special cases of our more general results. Because
of an explicit formula for the coefficients in terms of falling factorials established in ,
the polynomials A n (x; \Gammak) which we consider here are actually translates of the standard
higher order Bernoulli polynomials B (!)
n (x), but all of our significant results apply equally
well to the standard polynomials and their ordinary coefficients, with ! = n \Gamma k + 1. The
main results are summarized using standard notations in the research announcement .
Our approach differs from the usual one in that we make no use of congruence prop
erties of the Bernoulli numbers, and in particular do not use the von Staudt--Clausen
Theorem, which is an essential ingredient of the usual approach. Instead we characterize
the behavior of the p--adic poles of the coefficients in terms of the base p expansion of n.
The p--Eisenstein situation occurs when the highest order pole is simple.
The Bernoulli polynomials B n (x) are defined by (cf. [11, 12])