Faulhaber's formula

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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 .

The result: Faulhaber's formula[edit]

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.[1]

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

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.

A rigorous proof of these formulas and Faulhaber's assertion that such formulas would exist for all odd powers took until Carl Jacobi (1834), two centuries later.

Proof with exponential generating function[edit]


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 . This may be converted to a generating function with the convention by the addition of to the coefficient of in each ( does not need to be changed):
It follows immediately that
for all .

Faulhaber polynomials[edit]

The term Faulhaber polynomials is used by some authors to refer to another polynomial sequence related to that given above.


Faulhaber observed that if p is odd then is a polynomial function of a.

Proof without words for p = 3 [2]

For p = 1, it is clear that

For p = 3, the result that
is known as Nicomachus's theorem.

Further, we have

(see OEISA000537, OEISA000539, OEISA000541, OEISA007487, OEISA123095).

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 odd j > 1.

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.

Matrix form[edit]

Faulhaber's formula can also be written in a form using matrix multiplication.

Take the first seven examples

Writing these polynomials as a product between matrices gives

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 be the matrix obtained from by changing the signs of the entries in odd diagonals, that is by replacing by , let be the matrix obtained from with a similar transformation, then

This is because it is evident that and that therefore polynomials of degree of the form subtracted the monomial difference they become .

This is true for every order, that is, for each positive integer m, one has and 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.[3][4]


  • Replacing with , we find the alternative expression:
  • Subtracting from both sides of the original formula and incrementing by , we get
where can be interpreted as "negative" Bernoulli numbers with .
  • We may also expand in terms of the Bernoulli polynomials to find
    which implies
    Since whenever is odd, the factor may be removed when .
  • It can also be expressed in terms of Stirling numbers of the second kind and falling factorials as[5]
    This is due to the definition of the Stirling numbers of the second kind as mononomials in terms of falling factorials, and the behaviour of falling factorials under the indefinite sum.
  • There is also a similar (but somehow simpler) expression: using the idea of telescoping and the binomial theorem, one gets Pascal's identity:[6]

This in particular yields the examples below – e.g., take k = 1 to get the first example. In a similar fashion we also find

  • Faulhaber's formula was generalized by Guo and Zeng to a q-analog.[7]

Relationship to Riemann zeta function[edit]

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[edit]

In the umbral calculus, one treats the Bernoulli numbers , , , … as if the index j in were actually an exponent, and so as if the Bernoulli numbers were powers of some object B.

Using this notation, Faulhaber's formula can be written as

Here, the expression on the right must be understood by expanding out to get terms that can then be interpreted as the Bernoulli numbers. Specifically, using the binomial theorem, we get

A derivation of Faulhaber's formula using the umbral form is available in The Book of Numbers by John Horton Conway and Richard K. Guy.[8]

Classically, this umbral form was considered as a notational convenience. In the modern umbral calculus, on the other hand, this is given a formal mathematical underpinning. One considers the linear functional T on the vector space of polynomials in a variable b given by Then one can say

See also[edit]


  1. ^ Donald E. Knuth (1993). "Johann Faulhaber and sums of powers". Mathematics of Computation. 61 (203): 277–294. arXiv:math.CA/9207222. 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. ^ Gulley, Ned (March 4, 2010), Shure, Loren (ed.), Nicomachus's Theorem, Matlab Central
  3. ^ 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).
  4. ^ Derby, Nigel (2015), "A search for sums of powers", The Mathematical Gazette, 99 (546): 416–421, doi:10.1017/mag.2015.77.
  5. ^ Concrete Mathematics, 1st ed. (1989), p. 275.
  6. ^ Kieren MacMillan, Jonathan Sondow (2011). "Proofs of power sum and binomial coefficient congruences via Pascal's identity". American Mathematical Monthly. 118 (6): 549–551. arXiv:1011.0076. doi:10.4169/amer.math.monthly.118.06.549.
  7. ^ Guo, Victor J. W.; Zeng, Jiang (30 August 2005). "A q-Analogue of Faulhaber's Formula for Sums of Powers". The Electronic Journal of Combinatorics. 11 (2). arXiv:math/0501441. Bibcode:2005math......1441G. doi:10.37236/1876. S2CID 10467873.
  8. ^ John H. Conway, Richard Guy (1996). The Book of Numbers. Springer. p. 107. ISBN 0-387-97993-X.

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