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 as a polynomial in n. In modern notation, Faulhaber's formula is Here, is the binomial coefficient "p + 1 choose k", 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 as a (p + 1)th-degree polynomial function of n. The first few examples are well known. For p = 0, we have For p = 1, we have the triangular numbers For p = 2, we have the square pyramidal numbers The coefficients of Faulhaber's formula in its general form involve the Bernoulli numbers Bj. 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 Then Faulhaber's formula is that Here, the Bj are the Bernoulli numbers as above, and is the binomial coefficient "p + 1 choose k". So, for example, one has for p = 4, The first seven examples of Faulhaber's formula are 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 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.

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