Euler polynomial
In mathematics, the Bernoulli polynomials, named after Jacob Bernoulli, occur in the study of many special functions and in particular the Riemann zeta function and the Hurwitz zeta function. This is in large part because they are an Appell sequence, i.e. a Sheffer sequence for the ordinary derivative operator. Unlike orthogonal polynomials, the Bernoulli polynomials are remarkable in that the number of crossings of the x-axis in the unit interval does not go up as the degree of the polynomials goes up. In the limit of large degree, the Bernoulli polynomials, appropriately scaled, approach the sine and cosine functions.
The Bernoulli polynomials Bn admit a variety of different representations. Which among them should be taken to be the definition may depend on one's purposes.
for n ≥ 0, where bk are the Bernoulli numbers.
The generating function for the Bernoulli polynomials is
The generating function for the Euler polynomials is
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.
The Bernoulli polynomials are the unique polynomials determined by
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
Note the remarkable similarity to the globally convergent series expression for the Hurwitz zeta function. Indeed, one has
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