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Bernoulli polynomials and numbers∗
pahio†
2013-03-22 0:50:58
For n = 0, 1, 2, . . ., the Bernoulli polynomial may be defined as the uniquely
determined polynomial bn (x) satisfying
Z
x+1
bn (t) dt = xn .
(1)
x
The constant term of bn (x) is the nth Bernoulli number Bn .
The Bernoulli polynomial is often denoted also Bn (x).
The uniqueness of the solution bn (x) in (1) is justificated by the
Lemma. For any polynomial f (x), there exists a unique polynomial g(x)
with the same degree satisfying
Z x+1
g(t) dt = f (x).
(2)
x
Proof. For every n = 0, 1, 2, . . ., the polynomial
Z x+1
(x+1)n+1 − xn+1
gn (x) =:
tn dt =
n+1
x
is monic and its degree is n. If the coefficient of xn in f (x) is a0 , then the
difference f (x)−a0 gn (x) is a polynomial of degree n−1. Correspondingly we
obtain f (x) − a0 gn (x) − a1 gn−1 (x) having the degree n−2 and so on. Finally
we see that
f (x) − a0 gn (x) − a1 gn−1 (x) − . . . − an g0 (x)
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must be the zero polynomial. Therefore
f (x) = a0 gn (x) + a1 gn−1 (x) + . . . + an g0 (x)
n
X
=
ai gn−i (x)
i=0
=
n
X
Z
x+1
ai
tn−i dt
x
i=0
n
x+1 X
Z
=
x
whence we have g(x) =
n
X
ai tn−i dt
i=0
ai xn−i .
i=0
The proof implies also that the coefficients of g(x) are rational, if the coefficients of f (x) are such. So we know that all Bernoulli polynomials have only
rational coefficients.
The relation (1) implies easily, that the Bernoulli polynomials form an Appell
sequence.
References
[1] М. М. Постников:
Введение в теорию
Издательство “Наука”. Москва (1982).
алгебраических
чисел.
English translation:
M. M. Postnikov: Introduction to algebraic number theory. Science Publs
(“Nauka”). Moscow (1982).
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