Bernoulli polynomials

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.

The Bernoulli polynomials Bn can be defined by a generating function.

They also admit a variety of derived representations.

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

In,[1][2] it is deduced and proved that the Bernoulli polynomials can be obtained by the following integral recurrence

That is similar to the series expression for the Hurwitz zeta function in the complex plane.

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

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

An explicit formula for the Euler polynomials is given by

Lehmer (1940)[3] showed that the maximum value (Mn) of

is the Riemann zeta function), while the minimum (mn) obeys

The Bernoulli and Euler polynomials obey many relations from umbral calculus:

(Δ is the forward difference operator).

Zhi-Wei Sun and Hao Pan [4] established the following surprising symmetry relation: If r + s + t = n and x + y + z = 1, then

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

The Fourier series of the Euler polynomials may also be calculated.

Specifically, evidently from the above section on integral operators, it follows that

The Bernoulli polynomials may be expanded in terms of the falling factorial

denotes the Stirling number of the second kind.

The above may be inverted to express the falling factorial in terms of the Bernoulli polynomials:

denotes the Stirling number of the first kind.

The multiplication theorems were given by Joseph Ludwig Raabe in 1851: For a natural number m≥1,

Two definite integrals relating the Bernoulli and Euler polynomials to the Bernoulli and Euler numbers are:[5] Another integral formula states[6] with the special case for

A periodic Bernoulli polynomial Pn(x) is a Bernoulli polynomial evaluated at the fractional part of the argument x.

These functions are used to provide the remainder term in the Euler–Maclaurin formula relating sums to integrals.