Faulhaber's formula


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.

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 coincide with the first Bernoulli numbers denoted or simply except for j=1 being opposite of. 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:

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.
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
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:
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

They are polynomials in function of a variable which, when the variable coincides with the number of summands, calculate the sum of powers with bases in arithmetic progression and an exponent equal to the number preceding its degree. The problem is therefore to find polynomials such that:
with variable and parameters of the polynomial ; and non-negative integers, first term of an arithmetic progression and difference of the same progression, where and are any real or complex number
are the polynomials identified by the Faulhaber's formula presented posthumously by Jacob Bernoulli in 1713;
are the polynomials differing from the previous ones only in the sign of a monomial of degree 'p';
are the polynomials by sums of powers of successive odd numbers.

Matrix method

For any positive integer, the general case is solved using the following formula:
with and
where, and are integers.

Example

The formula in the particular case   becomes:  
Calculating the matrix T, whose elements follow the binomial theorem with the assigned values, that is, T, and finding the inverse matrix of the lower triangular matrix A obtained from the Pascal's triangle without the last element of each row, we have:
Multiplying the rows by the columns of the two matrices, we obtain:
and therefore:
Finally, if you are interested in the sums of the first three addends

Bernoulli's polynomial method

The following formula implicitly solves the problem using Bernoulli polynomials:
In particular:

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
.
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, 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 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.. 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 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. Beardon has published formulas for powers of, including a 1996 paper which demonstrated that integer powers of can be written as a linear sum of terms in the sequence :
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. A 2019 paper by Derby proved that:

This can be calculated in matrix form, as described below. 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 Maclaurin coefficients of — 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.

Matrix form

Faulhaber's formula can also be written in a form using matrix multiplication.
Take the first seven examples
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, 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.

Variations

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.

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.
Faulhaber's formula can be written in terms of the Hurwitz zeta function:

Umbral form

In the umbral calculus, one treats the Bernoulli numbers,,,... as if the index 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
A derivation of Faulhaber's formula using the umbral form is available in The Book of Numbers by John Horton Conway and Richard K. Guy.
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 functional on the vector space of polynomials in a variable given by Then one can say