↑ arvau

Wikipedia / 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 (i.e. a Sheffer sequence for the ordinary derivative operator). 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 exponential generating function

Explicit formula

are the Bernoulli numbers are the Euler numbers.

Representation by a differential operator

is differentiation with respect to and the fraction is expanded as a formal power series cf.

Representation by an integral operator

The integral transform on polynomials f This can be used to produce the inversion formulae below.

Integral recurrence

In,

Another explicit formula

That is similar to the series expression for the Hurwitz zeta function in the complex plane is the Hurwitz zeta function. The latter generalizes the Bernoulli polynomials, allowing for non-integer values

The inner sum may be understood to be the th forward difference is the forward difference operator

This formula may be derived from an identity appearing above as follows. Since the forward difference operator where is differentiation with respect to , we have, from the Mercator series

As long as this operates on an one may let go from only up

An integral representation for the Bernoulli polynomials is given by the Nörlund–Rice integral, which follows from the expression as a finite difference.

Sums of pth powers

Using either the above integral representation or the identity (assuming 00=1).

Explicit expressions for low degrees

Maximum and minimum

At higher showed that the maximum value ( between and unless is is the Riemann zeta function), while the minimum ( unless

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 ( is the forward difference operator These polynomial sequences are Appell sequences

Translations

These identities are also equivalent to saying that these polynomial sequences are Appell sequences. (Hermite polynomials are another example.)

Symmetries

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

Fourier series

The Fourier series of the Bernoulli polynomials is also a Dirichlet series 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 when and is valid for when .

They are related to the Legendre chi function

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

Relation to falling factorial

The Bernoulli polynomials may be expanded in terms of the falling factorial denotes the Stirling number of the second kind 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:

Another integral formula states

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.

:

See also

References

  • Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, (1972) Dover, New York. (See Chapter 23)

  • (See chapter 12.11)

  • (Reviews relationship to the Hurwitz zeta function and Lerch transcendent.)