Bernoulli polynomials


In mathematics, the Bernoulli polynomials, named after Jacob Bernoulli, combine the Bernoulli numbers and binomial coefficients. They are used for series expansion of functions, and with the Euler–MacLaurin formula.
These polynomials occur in the study of many special functions and, in particular the Riemann zeta function and the Hurwitz zeta function. They are an Appell sequence. For the Bernoulli polynomials, the number of crossings of the x-axis in the unit interval does not go up with the degree. In the limit of large degree, they approach, when appropriately scaled, the sine and cosine functions.
A similar set of polynomials, based on a generating function, is the family of Euler polynomials.

Representations

The Bernoulli polynomials Bn can be defined by a generating function. They also admit a variety of derived representations.

Generating functions

The generating function for the Bernoulli polynomials is
The generating function for the Euler polynomials is

Explicit formula

for n ≥ 0, where Bk are the Bernoulli numbers, and Ek are the Euler numbers.

Representation by a differential operator

The Bernoulli polynomials are also given by
where D = d/dx is differentiation with respect to x and the fraction is expanded as a formal power series. It follows that
cf. integrals below. By the same token, the Euler polynomials are given by

Representation by an integral operator

The Bernoulli polynomials are also the unique polynomials determined by
The integral transform
on polynomials f, simply amounts to
This can be used to produce the [|inversion formulae below].

Another explicit formula

An explicit formula for the Bernoulli polynomials is given by
That is similar to the series expression for the Hurwitz zeta function in the complex plane. Indeed, there is the relationship
where ζ is the Hurwitz zeta function. The latter generalizes the Bernoulli polynomials, allowing for non-integer values of n.
The inner sum may be understood to be the nth forward difference of xm; that is,
where Δ is the forward difference operator. Thus, one may write
This formula may be derived from an [|identity] appearing above as follows. Since the forward difference operator Δ equals
where D is differentiation with respect to x, we have, from the Mercator series,
As long as this operates on an mth-degree polynomial such as xm, one may let n go from 0 only up to m.
An [|integral representation] for the Bernoulli polynomials is given by the Nörlund-Rice integral, which follows from the expression as a finite difference.
An explicit formula for the Euler polynomials is given by
The above follows analogously, using the fact that

Sums of ''p''th powers

Using either the above integral representation of or the identity, we have
. See Faulhaber's formula for more on this.

The Bernoulli and Euler numbers

The Bernoulli numbers are given by
This definition gives for.
An alternate convention defines the Bernoulli numbers as
The two conventions differ only for since.
The Euler numbers are given by

Explicit expressions for low degrees

The first few Bernoulli polynomials are:
The first few Euler polynomials are:

Maximum and minimum

At higher n, the amount of variation in Bn between x = 0 and x = 1 gets large. For instance,
which shows that the value at x = 0 is −3617/510 ≈ −7.09, while at x = 1/2, the value is 118518239/3342336 ≈ +7.09. D.H. Lehmer showed that the maximum value of Bn between 0 and 1 obeys
unless n is 2 modulo 4, in which case
, while the minimum obeys
unless n is 0 modulo 4, in which case
These limits are quite close to the actual maximum and minimum, and Lehmer gives more accurate limits as well.

Differences and derivatives

The Bernoulli and Euler polynomials obey many relations from umbral calculus:
. Also,
These polynomial sequences are Appell sequences:

Translations

These identities are also equivalent to saying that these polynomial sequences are Appell sequences.

Symmetries

Zhi-Wei Sun and Hao Pan established the following surprising symmetry relation: If and, then
where

Fourier series

The Fourier series of the Bernoulli polynomials is also a Dirichlet series, given by the expansion
Note the simple large n limit to suitably scaled trigonometric functions.
This is a special case of the analogous form for the Hurwitz zeta function
This expansion is valid only for 0 ≤ x ≤ 1 when n ≥ 2 and is valid for 0 < x < 1 when n = 1.
The Fourier series of the Euler polynomials may also be calculated. Defining the functions
and
for, the Euler polynomial has the Fourier series
and
Note that the and are odd and even, respectively:
and
They are related to the Legendre chi function as
and

Inversion

The Bernoulli and Euler polynomials may be inverted to express the monomial in terms of the polynomials.
Specifically, evidently from the above section on [|integral operators], it follows that
and

Relation to falling factorial

The Bernoulli polynomials may be expanded in terms of the falling factorial as
where and
denotes the Stirling number of the second kind. The above may be inverted to express the falling factorial in terms of the Bernoulli polynomials:
where
denotes the Stirling number of the first kind.

Multiplication theorems

The multiplication theorems were given by Joseph Ludwig Raabe in 1851:
For a natural number,

Integrals

Two definite integrals relating the Bernoulli and Euler polynomials to the Bernoulli and Euler numbers are:

Periodic Bernoulli polynomials

A periodic Bernoulli polynomial is a Bernoulli polynomial evaluated at the fractional part of the argument. These functions are used to provide the remainder term in the Euler–Maclaurin formula relating sums to integrals. The first polynomial is a sawtooth function.
Strictly these functions are not polynomials at all and more properly should be termed the periodic Bernoulli functions, and is not even a function, being the derivative of a sawtooth and so a Dirac comb.
The following properties are of interest, valid for all :

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