In mathematics, Faulhaber's formula, named after Johann Faulhaber, expresses the sum For other meanings of mathematics or math, see mathematics (disambiguation). ...

as a (p + 1)th-degree polynomial function of x, the coefficients involving Bernoulli numbers. In mathematics, a polynomial is an expression in which constants and variables are combined using only addition, subtraction, multiplication, and positive whole number exponents (raising to a power). ... In mathematics, the Bernoulli numbers Bn were first discovered in connection with the closed forms of the sums for various fixed values of n. ...

Note: By the most usual convention, the Bernoulli numbers are

But for the moment we follow a convention seen less often, that B₁ = +1/2, and all the other Bernoulli numbers remain as above (but see below for more on this).

The formula says

(the index j runs only up to p, not up to p + 1).

Faulhaber did not know the formula in this form. He did know at least the first 17 cases and the fact that when the exponent is odd, then the sum is a polynomial function of the sum in the special case that the exponent is 1. He also knew some remarkable generalizations (see Knuth).

The first several cases

Another form

One may see the formula stated with terms running from 1 to x − 1 rather than from 1 to x. In that case, the only thing that changes is that we take B₁ = −1/2 rather the +1/2, so that term of second-highest degree in each case has a minus sign rather than a plus sign.

Relation to Bernoulli polynomials

One may also write

where φ_j is the jth Bernoulli polynomial. In mathematics, the Bernoulli polynomials occur in the study of many special functions and in particular the Riemann zeta function and the Hurwitz zeta function. ...

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 In mathematics, before the 1970s, the term umbral calculus was understood to mean the surprising similarities between otherwise unrelated polynomial equations, and certain shadowy techniques that can be used to prove them. ... In mathematics, the binomial theorem is an important formula giving the expansion of powers of sums. ...

In the modern umbral calculus, one considers the linear functional T on the vector space of polynomials in a variable b given by In linear algebra, a branch of mathematics, a linear functional or linear form is a linear function from a vector space to its field of scalars. ... In mathematics, a vector space (or linear space) is a collection of objects (known as vectors) which may be scaled and added; all linear combinations of vectors are themselves vectors. ...

Then one can say

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

Some authors call these polynomials in y "Faulhaber polynomials".

References and external links

• The Book of Numbers, John H. Conway, Richard Guy, Spring, 1998, ISBN 038797993X, page 107
• CRC Concise Encyclopedia of Mathematics, Eric Weisstein, Chapman & Hall/CRC, 2003, ISBN 1584883472, page 2331
• "Johann Faulhaber and Sums of Powers" by Donald Knuth
• Faulhaber's formula on MathWorld
• "Darinnen die miraculosische Inventiones zu den höchsten Cossen weiters continuirt und profitiert werden", Academia Algebrae, Johann Faulhaber, Augpurg, bey Johann Ulrich Schöigs, 1631. Call number QA154.8 F3 1631a f MATH at Stanford University Libraries.