The formula says
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 (see History section below). The derivation of Faulhaber's formula is available in The Book of Numbers by John Horton Conway and Richard K. Guy.
This in particular yields the examples below, e.g., take k = 1 to get the first example.
- (the triangular numbers)
- (the square pyramidal numbers)
denote the sum under consideration for integer
Define the following exponential generating function with (initially) indeterminant
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 (with the convention ). 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.
By relabelling we find the alternative expression
We may also expand in terms of the Bernoulli polynomials to find
Relationship to Riemann Zeta Function
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 .
Then one can say
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
The first of these identities, for the case p = 3, is known as Nicomachus's theorem. Some authors call the polynomials on the right hand sides of these identities "Faulhaber polynomials in a". The polynomials in the right-hand sides are divisible by a 2 because for j > 1 odd the Bernoulli number Bj is 0.
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 to a.
Since a = n(n + 1)/2, these formulae show that for an odd power (greater than 1), the sum is a polynomial in n having factors n2 and (n + 1)2, while for an even power the polynomial has factors n, n + ½ and n + 1.
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 above.
- John H. Conway, Richard Guy (1996). The Book of Numbers. Springer. p. 107. ISBN 0-387-97993-X.
- Kieren MacMillan, Jonathan Sondow (2011). "Proofs of power sum and binomial coefficient congruences via Pascal's identity". American Mathematical Monthly 118: 549–551. doi:10.4169/amer.math.monthly.118.06.549.
- Donald E. Knuth (1993). "Johann Faulhaber and sums of powers". Math. Comp. (American Mathematical Society) 61 (203): 277–294. arXiv:math.CA/9207222. doi:10.2307/2152953. JSTOR 2152953. The arxiv.org paper has a misprint in the formula for the sum of 11th powers, which was corrected in the printed version. Correct version.
- Jacobi, Carl (1834). "De usu legitimo formulae summatoriae Maclaurinianae". Journal für die reine und angewandte Mathematik 12. pp. 263–72.
- Weisstein, Eric W., "Faulhaber's formula", MathWorld.
- Johann Faulhaber (1631). Academia Algebrae - Darinnen die miraculosische Inventiones zu den höchsten Cossen weiters continuirt und profitiert werden. A very rare book, but Knuth has placed a photocopy in the Stanford library, call number QA154.8 F3 1631a f MATH. (online copy at Google Books)
- Beardon, A. F. "Sums of Powers of Integers". MAA. Retrieved 2011-10-23. (winner of a 1997 Lester R. Ford Award)