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Summary: IRREDUCIBLE FACTORS AND P--ADIC POLES OF
HIGHER ORDER BERNOULLI POLYNOMIALS
ARNOLD ADELBERG
ABSTRACT. We establish the p--adic singularity pattern of the coef
ficients of the higher order Bernoulli polynomials, and use this to de
termine all instances of p--Eisenstein behavior. This approach provides
new proofs for and generalizes known irreducibility results.
1. INTRODUCTION
This announcement summarizes the author's most significant results on higher order
Bernoulli polynomials. Details of the proofs will appear in [2]. In this paper, we will use
standard notations rather than the terminology of [1] and [2].
The Bernoulli polynomials B n (x) are defined by [11,12]
te xt
e t \Gamma 1 =
1
X
n=0
B n (x) t n
n!
;
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