In mathematics, Faulhaber's formula, named after the early 17th century mathematician Johann Faulhaber, expresses the sum of the p-th powers of the first n positive integers
as a polynomial in n. In modern notation, Faulhaber's formula is
The coefficients of Faulhaber's formula in its general form involve the Bernoulli numbersBj. The Bernoulli numbers begin
where here we use the convention that . The Bernoulli numbers have various definitions (see Bernoulli number#Definitions), such as that they are the coefficients of the exponential generating function
Faulhaber's formula is also called Bernoulli's formula. Faulhaber did not know the properties of the coefficients later 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.[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
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.
A rigorous proof of these formulas and Faulhaber's assertion that such formulas would exist for all odd powers took until Carl Jacobi (1834), two centuries later.
where denotes the Bernoulli number with the convention . This may be converted to a generating function with the convention by the addition of to the coefficient of in each ( does not need to be changed):
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 numberBj is 0 for odd j > 1.
Inversely, writing for simplicity , we have
and generally
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 + 1/2 and n + 1.
Expressing products of power sums as linear combinations of power sumsedit
Products of two (and thus by iteration, several) power sums can be written as linear combinations of power sums with either all degrees even or all degrees odd, depending on the total degree of the product as a polynomial in , e.g. .
Note that the sums of coefficients must be equal on both sides, as can be seen by putting , which makes all the equal to 1. Some general formulae include:
Note that in the second formula, for even the term corresponding to is different from the other terms in the sum, while for odd , this additional term vanishes because of .
Writing these polynomials as a product between matrices gives
where
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 row, and with alternating signs.
Let be the matrix obtained from by changing the signs of the entries in odd diagonals, that is by replacing by , let be the matrix obtained from with a similar transformation, then
and
Also
This is because it is evident that
and that therefore polynomials of degree of the form subtracted the monomial difference they become .
This is true for every order, that is, for each positive integer m, one has and
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.[3][4]
This is due to the definition of the Stirling numbers of the second kind as mononomials in terms of falling factorials, and the behaviour of falling factorials under the indefinite sum.
Interpreting the Stirling numbers of the second kind, , as the number of set partitions of into parts, the identity has a direct combinatorial proof since both sides count the number of functions with maximal. The index of summation on the left hand side represents , while the index on the right hand side is represents the number of elements in the image of f.
There is also a similar (but somehow simpler) expression: using the idea of telescoping and the binomial theorem, one gets Pascal's identity:[6]
This in particular yields the examples below – e.g., take k = 1 to get the first example. In a similar fashion we also find
Faulhaber's formula was generalized by Guo and Zeng to a q-analog.[7]
In the umbral calculus, one treats the Bernoulli numbers , , , ... as if the index j in were actually an exponent, and so as if the Bernoulli numbers were powers of some object B.
Using this notation, Faulhaber's formula can be written as
Here, the expression on the right must be understood by expanding out to get terms that can then be interpreted as the Bernoulli numbers. Specifically, using the binomial theorem, we get
Classically, this umbral form was considered as a notational convenience. In the modern umbral calculus, on the other hand, this is given a formal mathematical underpinning. One considers the linear functionalT on the vector space of polynomials in a variable b given by Then one can say
The series as a function of m is often abbreviated as . Beardon (see External Links) have published formulas for powers of . For example, Beardon 1996 stated this general formula for powers of , which shows that raised to a power N can be written as a linear sum of terms ... For example, by taking N to be 2, then 3, then 4 in Beardon's formula we get the identities .
Other formulae, such as and are known but no general formula for , where m, N are positive integers, has been published to date. In an unpublished paper by Derby (2019) [9] the following formula was stated and proved:
.
This can be calculated in matrix form, as described above. In the case when m = 1 it replicates Beardon's formula for . When m = 2 and N = 2 or 3 it generates the given formulas for and . Examples of calculations for higher indices are
and .
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)