Faulhaber's formula

In mathematics, Faulhaber's formula, named after Johann Faulhaber, expresses the sum

$\sum_{k=1}^n k^p = 1^p + 2^p + 3^p + \cdots + n^p$

as a (p + 1)th-degree polynomial function of n, the coefficients involving Bernoulli numbers B_j.

The formula says

$\sum_{k=1}^n k^p = {1 \over p+1} \sum_{j=0}^p (-1)^j{p+1 \choose j} B_j n^{p+1-j},\qquad \mbox{where}~B_1 = -\frac{1}{2}.$

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 below). The derivation of Faulhaber's formula is available in The Book of Numbers by John Horton Conway and Richard K. Guy.

Examples

$1 + 2 + 3 + \cdots + n = {n(n+1) \over 2} = {n^2 + n \over 2}$ (the triangular numbers)

$1^2 + 2^2 + 3^2 + \cdots + n^2 = {n(n+1)(2n+1) \over 6} = {2n^3 + 3n^2 + n \over 6}$ (the square pyramidal numbers)

$1^3 + 2^3 + 3^3 + \cdots + n^3 = \left({n^2 + n \over 2}\right)^2 = {n^4 + 2n^3 + n^2 \over 4}$ (the squared triangular numbers)

$1^4 + 2^4 + 3^4 + \cdots + n^4 = {6n^5 + 15n^4 + 10n^3 - n \over 30}$

$1^5 + 2^5 + 3^5 + \cdots + n^5 = {2n^6 + 6n^5 + 5n^4 - n^2 \over 12}$

$1^6 + 2^6 + 3^6 + \cdots + n^6 = {6n^7 + 21n^6 + 21n^5 -7n^3 + n \over 42}$

Relation to Bernoulli polynomials

One may also write

$\sum_{k=0}^{n} k^p = \frac{\varphi_{p+1}(n+1)-\varphi_{p+1}(0)}{p+1},$

where φ_j is the jth Bernoulli polynomial.

Umbral form

In the classic umbral calculus one formally treats the indices j in a sequence B_j as if they were exponents, so that, in this case we can apply the binomial theorem and say

$\sum_{k=1}^n k^p = {1 \over p+1} \sum_{j=0}^p {p+1 \choose j} B_j n^{p+1-j} = {1 \over p+1} \sum_{j=0}^p {p+1 \choose j} B^j n^{p+1-j}$

$= {(B+n)^{p+1} - B^{p+1} \over p+1}.$

In the modern umbral calculus, one considers the linear functional T on the vector space of polynomials in a variable b given by

$T(b^j) = B_j.\,$

Then one can say

$\sum_{k=1}^n k^p = {1 \over p+1} \sum_{j=0}^p {p+1 \choose j} B_j n^{p+1-j} = {1 \over p+1} \sum_{j=0}^p {p+1 \choose j} T(b^j) n^{p+1-j}$

$= {1 \over p+1} T\left(\sum_{j=0}^p {p+1 \choose j} b^j n^{p+1-j} \right) = T\left({(b+n)^{p+1} - b^{p+1} \over p+1}\right).$

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

$1^p + 2^p + 3^p + \cdots + n^p$

is a polynomial function of

$a=1+2+3+\cdots+n= \frac{n(n+1)}{2}.$

In particular:

$1^3 + 2^3 + 3^3 + \cdots + n^3 = a^2;$  A000537

$1^5 + 2^5 + 3^5 + \cdots + n^5 = {4a^3 - a^2 \over 3};$  A000539

$1^7 + 2^7 + 3^7 + \cdots + n^7 = {12a^4 -8a^3 + 2a^2 \over 6};$  A000541

$1^9 + 2^9 + 3^9 + \cdots + n^9 = {16a^5 - 20a^4 +12a^3 - 3a^2 \over 5};$  A007487

$1^{11} + 2^{11} + 3^{11} + \cdots + n^{11} = {32a^6 - 64a^5 + 68a^4 - 40a^3 + 10a^2 \over 6}.$  A123095

More generally,

$\begin{align} 1^{2p+1} + 2^{2p+1} &+ 3^{2p+1} + \cdots + n^{2p+1}\\ &= \frac{1}{2^{2p+2}(2p+2)} \sum_{q=0}^p \binom{2p+2}{2q} (2-2^{2q})~ B_{2q} ~\left[(8a+1)^{p+1-q}-1\right]. \end{align}$

The first of these identities, for the case p = 3, is known as Nicomachus's theorem. Some authors call the polynomials on the right hand sides of these identities "Faulhaber polynomials in a". The polynomials in the right-hand sides are divisible by a ² because for j > 1 odd the Bernoulli number B_j is 0.

Faulhaber also knew that if a sum for an odd power is given by

$\sum_{k=1}^n k^{2m+1} = c_1 a^2 + c_2 a^3 + \cdots + c_m a^{m+1}$

then the sum for the even power just below is given by

$\sum_{k=1}^n k^{2m} = \frac{n+1/2}{2m+1}(2 c_1 a + 3 c_2 a^2+\cdots + (m+1) c_m a^m).$

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. As an application, the atomic numbers of every other alkaline earth metal (Be, Ca, Ba) are given by (4/3)n(n + 1/2)(n + 1)^{[citation needed]}.

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 above.

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

References and 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" from 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.. "Sums of Powers of Integers". MAA. http://mathdl.maa.org/images/upload_library/22/Ford/Beardon201-213.pdf. Retrieved 2011-10-23.