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 ("p" + 1)th-degree polynomial function of "n", the coefficients involving Bernoulli numbers "B", in the form submitted by Jacob Bernoulli and published in 1713: where formula_3 is a falling factorial. History. 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. A rigorous proof of these formulas and Faulhaber's assertion that such formulas would exist for all odd powers took until , two centuries later. Faulhaber polynomials. The term "Faulhaber polynomials" is used by some authors to refer to something other than the polynomial sequence given above. Faulhaber observed that if "p" is odd, then is a polynomial function of In particular: The first of these identities (the case "p" = 3) is known as Nicomachus's theorem. More generally, Some authors call the polynomials in "a" on the right-hand sides  of these identities Faulhaber polynomials. These polynomials are divisible by because the Bernoulli number is 0 for odd . 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 "n" and ("n" + 1), while for an even power the polynomial has  factors "n", "n" + ½ and "n" + 1. Section::::"Summae Potestatum". 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 formula_16, or using the Bernoulli number of the first kind for which formula_18 For example, as one has for , 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 (see History section). The derivation of Faulhaber's formula is available in "The Book of Numbers" by John Horton Conway and Richard K. Guy. 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 From examples to matrix theorem. From the previous examples we get : 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 formula_38 be the matrix obtained from formula_39 by changing the signs of the entries in odd diagonals, that is by replacing formula_40 by formula_41, let formula_42 be the matrix obtained from formula_43 with a similar transformation, then and also This is because it is evident that formula_46and that therefore polynomials of degree formula_47 of the form formula_48 subtracted the monomial difference formula_49 they become formula_50. This is  true for every order, that is, for each positive integer , one has formula_51 and formula_52 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. Proof with exponential generating function. Let denote the sum under consideration for integer formula_54 Define the following exponential generating function with (initially) indeterminate formula_55 We find This is an entire function in formula_55 so that formula_55 can be taken to be any complex number. We next recall  the exponential generating function for the Bernoulli polynomials formula_60 where formula_62 denotes the Bernoulli number with the convention formula_63. This may be converted to a generating function with the convention formula_64 by the addition of formula_65 to the coefficient of formula_66 in each formula_60 (formula_68 does not need to be changed): It follows immediately that for all formula_71. Alternate expressions. If we consider the generating function formula_73 in the large formula_83 limit for formula_84, then we find Heuristically, this suggests that This result agrees with the value of the Riemann zeta function formula_87 for negative integers formula_88 on appropriately analytically  continuing formula_89. Umbral form. In the classical umbral calculus one formally treats the indices "j" in a sequence "B" as if they were exponents, so that, always considering the variant formula_90, in this case we can apply the binomial theorem and say In the "modern" umbral calculus, one considers the linear functional "T" on the vector space of polynomials in a variable "b" given by Then one can say