In mathematics, Faulhaber's formula, named after Johann Faulhaber, expresses the sum of the p-th powers of the first n positive integers
as a (p + 1)th-degree polynomial function of n, the coefficients involving Bernoulli numbers Bj, in the form submitted by Jacob Bernoulli and published in 1713:
where is a falling factorial.
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 Carl Jacobi (1834).
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
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The first of these identities (the case p = 3) is known as Nicomachus's theorem.
Some authors call the polynomials in a on the right-hand sides of these identities Faulhaber polynomials. These polynomials are divisible by a2 because the Bernoulli number Bj is 0 for j > 1 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 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.
In 1713, Jacob Bernoulli published under the title Summae Potestatum an expression of the sum of the p powers of the n first integers as a (p + 1)th-degree polynomial function of n, with coefficients involving numbers Bj, now called Bernoulli numbers:
Introducing also the first two Bernoulli numbers (which Bernoulli did not), 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 p = 4,
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). 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 (but somehow simpler) expression: using the idea of telescoping and the binomial theorem, one gets Pascal's identity:
This in particular yields the examples below – e.g., take k = 1 to get the first example.
- (the triangular numbers)
- (the square pyramidal numbers)
- (the triangular numbers squared)
From examples to matrix theorem
From the previous examples we get:
- Writing these polynomials as a product between matrices gives
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 triangular Pascal 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 m, 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.
Proof with complex numbers
denote the sum under consideration for integer
Define the following exponential generating function with (initially) indeterminate
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
Since whenever is odd, the factor may be removed when .
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 .
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 functional T on the vector space of polynomials in a variable b given by
Then one can say