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. 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.