Faulhaber's formula

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In mathematics, Faulhaber's formula, named after 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 Bj, in the form submitted by Jacob Bernoulli and published in 1713:

where is a falling factorial.

History

Faulhaber's formula is also called Bernoulli's formula. Faulhaber did not know the properties of the coefficients 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.[1]

A rigorous proof of these formulas and his assertion that such formulas would exist for all odd powers took until Carl Jacobi (1834).

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:

OEISA000537


OEISA000539


OEISA000541


OEISA007487


OEISA123095

The first of these identities (the case p = 3) is known as Nicomachus's theorem.

More generally,[citation needed]

Some authors call the polynomials in a on the right-hand sides of these identities Faulhaber polynomials. These polynomials are divisible by a2 because the Bernoulli number Bj is 0 for j > 1 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 n2 and (n + 1)2, while for an even power the polynomial has factors n, n + ½ and n + 1.

Summae Potestatum

Jakob Bernoulli's Summae Potestatum, Ars Conjectandi, 1713

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

For example, as one has for p = 4,

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.[2]

There is also a similar (but somehow simpler) expression: using the idea of telescoping and the binomial theorem, one gets Pascal's identity:[3]

This in particular yields the examples below – e.g., take k = 1 to get the first example.

Examples

(the triangular numbers)
(the square pyramidal numbers)
(the squared triangular numbers)

From examples to matrix theorem

From the previous examples we get:

Writing these polynomials as a product between matrices gives

Surprisingly, inverting the matrix of polynomial coefficients yields something more familiar:

In the inverted matrix, the Pascal triangle is recognized but without the last element of each line and with alternate signs.

We define to be the Pascal triangle without the last element of each line:

Further, we define to be with alternate signs, which is with the following operation performed on each element: .

And we can conclude that has the property that .

As it has been discovered and proved,[4] this case is generalizable. For each positive integer m . 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 reversing the matrix easily obtained from the triangle of Pascal.

Similarly Nigel Derby in his paper[5] shows that the vector of the coefficients of Faulhaber polynomials is obtained by multiplying a complete line vector of the Pascal triangle for a triangular matrix extracted. For example in the case already considered:

where is like but with all positive elements and

Proof with complex numbers

Let

denote the sum under consideration for integer

Define the following exponential generating function with (initially) indeterminate

We find

This is an entire function in so that can be taken to be any complex number.

We next recall the exponential generating function for the Bernoulli polynomials

where denotes the Bernoulli number (with the convention ). We obtain the Faulhaber formula by expanding the generating function as follows:

Note that for all odd . Hence some authors define so that the alternating factor is absent.

Alternate expressions

By relabelling we find the alternative expression

We may also expand in terms of the Bernoulli polynomials to find

which implies

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 .

Umbral form

In the classical umbral calculus one formally treats the indices j in a sequence Bj as if they were exponents, so that, 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


Notes

  1. ^ Donald E. Knuth (1993). "Johann Faulhaber and sums of powers". Mathematics of Computation. 61 (203): 277–294. arXiv:math.CA/9207222Freely accessible. doi:10.2307/2152953. JSTOR 2152953.  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.
  2. ^ John H. Conway, Richard Guy (1996). The Book of Numbers. Springer. p. 107. ISBN 0-387-97993-X. 
  3. ^ Kieren MacMillan, Jonathan Sondow (2011). "Proofs of power sum and binomial coefficient congruences via Pascal's identity". American Mathematical Monthly. 118: 549–551. arXiv:1011.0076Freely accessible. doi:10.4169/amer.math.monthly.118.06.549. 
  4. ^ *Pietrocola, Giorgio (2017), On polynomials for the calculation of sums of powers of successive integers and Bernoulli numbers deduced from the Pascal’s triangle (PDF) .
  5. ^ *Derby, Nigel (2015), "A search for sums of powers", The Mathematical Gazette .

External links

  • Jacobi, Carl (1834). "De usu legitimo formulae summatoriae Maclaurinianae". Journal für die reine und angewandte Mathematik. 12. pp. 263–72. 
  • Weisstein, Eric W. "Faulhaber's formula". MathWorld. 
  • Johann Faulhaber (1631). Academia Algebrae - Darinnen die miraculosische Inventiones zu den höchsten Cossen weiters continuirt und profitiert werden.  A very rare book, but Knuth has placed a photocopy in the Stanford library, call number QA154.8 F3 1631a f MATH. (online copy at Google Books)
  • Beardon, A. F. (1996). "Sums of Powers of Integers" (PDF). American Mathematical Monthly. pp. 201–213. Retrieved 2011-10-23.  (Winner of a Lester R. Ford Award)
  • Schumacher, Raphael (2016). "An Extended Version of Faulhaber's Formula" (PDF). Journal of Integer Sequences. 19. 
  • Orosi, Greg (2018). "A Simple Derivation Of Faulhaber's Formula" (PDF). Applied Mathematics E-Notes. 18. pp. 124–126. 
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