Bernoulli polynomials

In mathematics, the Bernoulli polynomials 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 due to the fact that they are Sheffer sequences 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.

Fourier series

The Fourier series of the Bernoulli polynomials is also a Dirichlet series and is a special case of the Hurwitz zeta function

Related Topics:
Fourier series - Dirichlet series - Hurwitz zeta function

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:B_n(x) = -Gamma(n+1) sum_{k=1}^infty

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rac{ exp (2pi ikx) + exp (2pi ik(1-x)) } { (2pi ik)^n }.

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