Another Explicit Formula
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
where ζ(s, q) is the Hurwitz zeta; thus, in a certain sense, the Hurwitz zeta generalizes the Bernoulli polynomials to 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. 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 xm, one may let n go from 0 only up to m.
An integral representation for the Bernoulli polynomials is given by the Nörlund–Rice integral, which follows from the expression as a finite difference.
An explicit formula for the Euler polynomials is given by
This may also be written in terms of the Euler numbers Ek as
Read more about this topic: Bernoulli Polynomials
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