Faulhaber's formula

The Faulhabersche formula, named after Johannes Faulhaber by DE Knuth, describes, how can the sum of the first - th powers of a polynomial in charge of the degree.

The coefficients of the polynomial can thereby be calculated by means of Bernoulli numbers.

Representation of the polynomial by means of the Bernoulli numbers

For the calculation of the coefficients of this polynomial, the Bernoulli numbers are needed. In the following, denote the -th Bernoulli number of the first kind and the Bernoulli numbers of the second kind, then sees the Faulhabersche formula as follows:

If you look at the first place only the first powers, so you can see the Faulhabersche formula also " without exception" to describe and receives

Explicit representations

The low coefficient of unit fractions, as we know from the school mathematics at small k, but are not typical of the rest. Even at k = 11 is effective for the first time on a coefficient > 1; at even higher powers is the rule. This is due to the Bernoulli numbers which rise after a series of low values ​​strong, even stronger than any exponential function, and go to infinity. You yourself are the coefficients of the linear terms, and since they are at odd exponent equal to 1 zero, these members also lack accordingly in the sum formulas.

General:

Here you can see the beautiful context of Cavalieri's integral formula; a sum of powers is a power with a higher by 1 degree. This also applies to the special case of the trivial k = 0, because the integral of a function is a linear constant.

When expanding to first obtained in the divergent harmonic series, but in all convergent power sums. Your limits are, by definition, the function values ​​of the Riemann zeta function.

These are all special cases of the general Euler - Maclaurin formula applied to the function with any real is the exponent.

Related to Bernoulli polynomials

The sum of the first - th powers can also be expressed by means of Bernoulli polynomials:

Here, denotes the -th Bernoulli polynomial.

Faulhaber polynomials

The sums of odd powers

Can be represented as a polynomial in. Such polynomials in place in are also known as Faulhaber polynomials. John Faulhaber himself only knew the formula in the following form described, and calculated only the odd cases as a polynomial in and assumed that for all odd numbers exist a corresponding polynomial, but without giving a proof. The concept of the Bernoulli numbers he was unaware of.

Some examples of small exponent:

As a general rule for all:

Which is a polynomial of degree at or explicitly as a polynomial in

Historical

Faulhaber himself knew the formula in its present general form, not even the Bernoulli numbers were not known at the time. However, he knew at least the first 17 cases and the constructions of polynomials named after him. In 1834, Carl Gustav Jacob Jacobi published the first known proof of Faulhaber 's formula and used to the Euler - Maclaurin formula. More evidence has been published, among others, 1923 by L.Tits and 1986 by AWF Edwards. Donald Ervin Knuth studied generalizations of the sums representations as polynomials in fixed, and contributed to the popularization of Faulhaber 's polynomials in.