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Concept
Bernoulli polynomials
Summary
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.
The Bernoulli polynomials Bn can be defined by a generating function. They also admit a variety of derived representations.
The generating function for the Bernoulli polynomials is
The generating function for the Euler polynomials is
for n ≥ 0, where Bk are the Bernoulli numbers, and Ek are the Euler numbers.
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
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.
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(s, q) 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.
Official source
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