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.

Explicit formula

An explicit formula for the Bernoulli polynomials is given by

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:B_m(x)=

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sum_{n=0}^m rac{1}{n+1}

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sum_{k=0}^n (-1)^k {n choose k} (x+k)^m.

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Note the remarkable similarity to the globally convergent series expression for the Hurwitz zeta function. Indeed, one has

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:B_n(x) = -n zeta(1-n,x)

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where zeta(s,q) is the Hurwitz zeta; this, in a certain sense, the Hurwitz zeta generalizes the Bernoulli polynomials to non-integer values of n.

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An explicit formula for the Euler polynomials is given by

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:E_m(x)=

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sum_{n=0}^m rac{1}{2^n}

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sum_{k=0}^n (-1)^k {n choose k} (x+k)^m.

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