Faulhaber's formula is also called Bernoulli's formula. Faulhaber did not know the properties of the coefficients discovered by Bernoulli. Rather, he knew at least the first 17 cases, as well as the existence of the Faulhaber polynomials for odd powers described below. A rigorous proof of these formulas and his assertion that such formulas would exist for all odd powers took until.
Faulhaber polynomials
The term Faulhaber polynomials is used by some authors to refer to something other than the polynomial sequence given above. Faulhaber observed that if p is odd, then is a polynomial function of In particular: The first of these identities is known as Nicomachus's theorem. More generally, Some authors call the polynomials in a on the right-hand sides of these identities Faulhaber polynomials. These polynomials are divisible by because the Bernoulli number is 0 for odd. Faulhaber also knew that if a sum for an odd power is given by then the sum for the even power just below is given by Note that the polynomial in parentheses is the derivative of the polynomial above with respect toa. Since a = n/2, these formulae show that for an odd power, the sum is a polynomial in n having factorsn2 and 2, while for an even power the polynomial has factors n, n + ½ and n + 1.
''Summae Potestatum''
In 1713, Jacob Bernoulli published under the title Summae Potestatum an expression of the sum of the powers of the first integers as a th-degree polynomial function of , with coefficients involving numbers, now called Bernoulli numbers: Introducing also the first two Bernoulli numbers, the previous formula becomes using the Bernoulli number of the second kind for which, or using the Bernoulli number of the first kind for which For example, as one has for, Faulhaber himself did not know the formula in this form, but only computed the first seventeen polynomials; the general form was established with the discovery of the Bernoulli numbers. The derivation of Faulhaber's formula is available in The Book of Numbers by John Horton Conway and Richard K. Guy. There is also a similar expression: using the idea of telescoping and the binomial theorem, one getsPascal's identity: This in particular yields the examples below – e.g., take to get the first example.
Examples
From examples to matrix theorem
From the previous examples we get: Surprisingly, inverting the matrix of polynomial coefficients yields something more familiar: In the inverted matrix, Pascal's triangle can be recognized, without the last element of each line, and with alternate signs. More precisely, let be the lower triangularPascal matrix: Let be the matrix obtained from by changing the signs of the entries in odd diagonals, that is by replacing by. Then This is true for every order, that is, for each positive integer, one has Thus, it is possible to obtain the coefficients of the polynomials of the sums of powers of successive integers without resorting to the numbers of Bernoulli but by inverting the matrix easily obtained from the triangle of Pascal. One has also where is obtained from by removing the minus signs.
Let denote the sum under consideration for integer Define the following exponential generating function with indeterminate We find This is an entire function in so that can be taken to be any complex number. We next recall the exponential generating function for the Bernoulli polynomials where denotes the Bernoulli number. We obtain the Faulhaber formula by expanding the generating function as follows: Note that for all odd. Hence some authors define so that the alternating factor is absent.
Alternate expressions
By relabelling we find the alternative expression We may also expand in terms of the Bernoulli polynomials to find which implies Since whenever is odd, the factor may be removed when.
Using, one can write If we consider the generating function in the large limit for, then we find Heuristically, this suggests that This result agrees with the value of the Riemann zeta function for negative integers on appropriately analytically continuing.
Umbral form
In the classical umbral calculus one formally treats the indices j in a sequence Bj as if they were exponents, so that, in this case we can apply the binomial theorem and say In the modern umbral calculus, one considers the linear functionalT on the vector space of polynomials in a variable b given by Then one can say
