Faulhaber's formula
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
is the binomial coefficient "p + 1 choose r", and the Bj are the Bernoulli numbers with the convention that
Faulhaber's formula concerns expressing the sum of the p-th powers of the first n positive integers
The coefficients of Faulhaber's formula in its general form involve the Bernoulli numbers Bj.
The history of the problem begins in antiquity and coincides with that of some of its special cases.
This problem is quite simple but the case already known by the Pythagorean school for its connection with triangular numbers is historically interesting: For
of the cases, to which the two preceding polynomials belong, constitutes the classical problem of powers of successive integers.
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
Rather, he knew at least the first 17 cases, as well as the existence of the Faulhaber polynomials for odd powers described below.
Jacobi benefited from the progress of mathematical analysis using the development in infinite series of an exponential function generating Bernoulli numbers.
Edwards publishes an article [3] in which he shows that Pascal's identity can be expressed by means of triangular matrices containing the Pascal's triangle deprived of 'last element of each line: The example is limited by the choice of a fifth order matrix but is easily extendable to higher orders.
Other authors after Edwards dealing with various aspects of the power sum problem take the matrix path [6] and studying aspects of the problem in their articles useful tools such as the Vandermonde vector.
[7] Other researchers continue to explore through the traditional analytic route [8] and generalize the problem of the sum of successive integers to any geometric progression[9][10] Let
Define the following exponential generating function with (initially) indeterminate
We next recall the exponential generating function for the Bernoulli polynomials
Note that the sums of coefficients must be equal on both sides, as can be seen by putting
Faulhaber's formula can also be written in a form using matrix multiplication.
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.
by changing the signs of the entries in odd diagonals, that is by replacing
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.
parts, the identity has a direct combinatorial proof since both sides count the number of functions
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.
Faulhaber's formula can be written in terms of the Hurwitz zeta function:
A derivation of Faulhaber's formula using the umbral form is available in The Book of Numbers by John Horton Conway and Richard K.
In the modern umbral calculus, on the other hand, this is given a formal mathematical underpinning.
One considers the linear functional T on the vector space of polynomials in a variable b given by
A 2019 paper by Derby[19] proved that: This can be calculated in matrix form, as described above.
