In mathematics, Faulhaber's formula, named after the early 17th century mathematician Johann Faulhaber, expresses the sum of the th powers of the first positive integers
as a polynomial in . In modern notation, Faulhaber's formula is
Here, is the binomial coefficient " choose ", and the are the second Bernoulli numbers, identical to the first ones except for .With Bernoulli's first numbers, on the other hand, we have
The result: Faulhaber's formula
Faulhaber's formula concerns expressing the sum of the th powers of the first positive integers
as a th-degree polynomial function of .
The first few examples are well known. For , we have
For , we have the triangular numbers
For , we have the square pyramidal numbers
The coefficients of Faulhaber's formula in its general form involve the second Bernoulli numbers which nearly coincide with the first Bernoulli numbers denoted (or simply ); the sole exception is at , where but . The Bernoulli numbers begin
Then Faulhaber's formula is that
Here, the are the Bernoulli numbers as above, and
is the binomial coefficient " choose ".
Examples
So, for example, one has for ,
The first seven examples of Faulhaber's formula are
History Ancient period The history of the problem begins in antiquity, its special cases arising as solutions to related inquiries. The case coincides historically with the problem of calculating the sum of the first terms of an arithmetic progression. In chronological order, early discoveries include: , a formula known by the Pythagorean school for its connection with triangular numbers. a result showing that the sum of the first positive odd numbers is a perfect square. This formula was likely also known to the Pythagoreans, who in constructing figurate numbers realized that the gnomon of the th perfect square is precisely the th odd number. a formula that calculates the sum of the squares of the first positive integers, as demonstrated in Spirals, a work of Archimedes. a formula that calculates the sum of the cubes of the first positive integers, discovered as a corollary of a theorem of Nicomachus of Gerasa.
Middle period Over time, many other mathematicians became interested in the problem and made various contributions to its solution. These include Aryabhata, Al-Karaji, Ibn al-Haytham, Thomas Harriot, Johann Faulhaber, Pierre de Fermat and Blaise Pascal who recursively solved the problem of the sum of powers of successive integers by considering an identity that allowed to obtain a polynomial of degree already knowing the previous ones.
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. 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.
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 (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
A rigorous proof of these formulas and Faulhaber's assertion that such formulas would exist for all odd powers took until , two centuries later. Jacobi benefited from the progress of mathematical analysis using the development in infinite series of an exponential function generating Bernoulli numbers.
Modern period In 1982, A.W.F. Edwards published an article showing that Pascal's identity can be expressed by means of triangular matrices containing a modified Pascal's triangle:
The first element of the vector of the sums is and not because of the first addend, the indeterminate form , which should otherwise be assigned a value of 1
This example is limited by the choice of a fifth-order matrix, but the underlying method is easily extendable to higher orders. Writing the equation as and multiplying the two sides of the equation to the left by , we obtain , thereby arriving at the polynomial coefficients without directly using the Bernoulli numbers. Expanding on Edwards' work, some authors researching the power-sum problem have taken the matrix path, leveraging useful tools such as the Vandermonde vector. Other researchers continue to explore through the traditional analytic route, generalizing the problem of the sum of successive integers to any arithmetic progression.
Polynomials calculating sums of powers of arithmetic progressions Consider the problem of finding polynomials for any nonnegative integer such that
given complex numbers and . Faulhaber's formula handles the simple case . The polynomials compute sums of powers of successive odd numbers, and so on. In general, such polynomials exist for any arithmetic progression.
Matrix method The general case is solved using the following matrix formula: , with row-column definitions
where (row) and (column) are bound by a given matrix order .
Example To generalize up to , apply the above formula for matrix order :
Note that the nonzero elements of follow the binomial theorem, and that is just Pascal's triangle with each row's last element omitted. Letting , and computing , we have:
Multiplication therefore yields
Methods involving Bernoulli polynomials The following formula implicitly solves the problem using Bernoulli polynomials:
In particular:
Furthermore, given the lower triangular matrix
one has the substitution , allowing for a refinement of the above matrix method.
Faulhaber's formula in matrix form Using the matrix method with , Faulhaber's formula may be written as a product of matrices derived from Pascal's triangle. Letting , we have
choosing . This can be simplified given the surprising fact that
Let be the leftmost matrix. Then , and so . Hence
by substitution. Note the alternating signs in the inverted matrix. Since is the identity matrix, the equation for the alternatively indexed power sums follows readily:
Since these matrix equations hold for every order, it is possible to obtain the polynomials of the sums of powers of successive integers without resorting to the numbers of Bernoulli but by inverting a matrix easily obtained from the triangle of Pascal...
Faulhaber polynomials
The term Faulhaber polynomials is used by some authors to refer to another polynomial sequence related to that given above.
Write
Faulhaber observed that if is odd then is a polynomial function of .
For , it is clear that
For , the result that
is known as Nicomachus's theorem.
Further, we have
(see , , , , ).
More generally,
Some authors call the polynomials in on the right-hand sides of these identities Faulhaber polynomials. These polynomials are divisible by because the Bernoulli number is 0 for odd .
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 .
Since , these formulae show that for an odd power (greater than 1), the sum is a polynomial in having factors and , while for an even power the polynomial has factors , and .
Expressing products of power sums as linear combinations of power sums
Products of two (and thus by iteration, several) power sums with can be written as linear combinations of power sums with either all degrees even or all degrees odd, depending on the parity of the total degree of the product as a polynomial in , e.g. . The sums of coefficients on both sides must be equal, which follows by letting . Some general formulae include:
The latter formula may be used to recursively compute Faulhaber polynomials. Note that in the third formula, the quantity vanishes for odd since . Beardon has published formulas for powers of , including a 1996 paper which demonstrated that positive-integer powers of can be expressed as linear combinations of the even-degree power sums :
The first few resulting identities are then
.
Although other specific cases of – including and – are known, no explicit formula for for positive integers and has yet been reported.
Kariya showed that
which was generalized to arbitrary positive-integer powers by Derby:
.
The case replicates Beardon's formula for and confirms the above-stated results for and or . Results for higher powers include:
.
Further generalization is possible by considering the arbitrary power-sum product given positive integers . For convenience define , and let be the (possibly nonzero) coefficients of the polynomial — i.e. . It can be shown that
.
In particular, the product has trivially retrievable coefficients:
.
Combining with the above gives
which is an indexical restatement of Beardon's formula. More generally,
.
Compared to Derby's approach, this formula only requires knowledge of the coefficients of .
Recurrence relation between sums of powers Extending the Faulhaber's formula as a polynomial over the reals denoted by , it satisfies the first-order recurrence relation
The polynomial is the derivative of with respect to , and the constants are again the second Bernoulli numbers. This recurrence relation can be integrated to yield sums of higher powers:
These results can be seen as corollaries to the integral recurrence of the Bernoulli polynomials. This recurrence relation is easy to remember as it is much akin to the derivatives of polynomials: with an addition of the second Bernoulli numbers.
Calculated examples The first four calculated examples are as shown below: since
we have
In practice, when deriving from , one can simply determine by noting that .
Proof We know that is a polynomial. Since it coincides with the power sum at integer values, its extension to the reals should satisfy
Similarly, we have
Note that the latter is a multiple of away from the derivative of the former. Defining the residual , we have
I.e., is 1-periodic. By restricting the residual to be a polynomial, it must be a constant depending on . Thus, the recurrence relation is obtained.
Variations
Replacing with , we find the alternative expression: Subtracting from both sides of the original formula and incrementing by , we get where can be interpreted as "negative" Bernoulli numbers with . We may also expand in terms of the Bernoulli polynomials to find which implies Since whenever is odd, the factor may be removed when . It can also be expressed in terms of Stirling numbers of the second kind and falling factorials asConcrete Mathematics, 1st ed. (1989), p. 275. This is due to the definition of the Stirling numbers of the second kind as monomials 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 represents the number of elements in the image of . 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 to get the first example. In a similar fashion we also find
A generalized expression involving the Eulerian numbers is . Faulhaber's formula was generalized by Guo and Zeng to a -analog.
Relationship to Riemann zeta function
Using , one can write