Faulhaber's formula explained

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$$\backslash sum\_^n\; k^p\; =\; 1^p\; +\; 2^p\; +\; 3^p\; +\; \backslash cdots\; +\; n^p$$as a polynomial in n. In modern notation, Faulhaber's formula is$$\backslash sum\_^n\; k^\; =\; \backslash frac\; \backslash sum\_^p\; \backslash binom\; B\_r\; n^\; .$$Here, $\backslash binom$ is the binomial coefficient "p + 1 choose r", and the B_j are the Bernoulli numbers with the convention that $B\_1\; =\; +\backslash frac12$.

The result: Faulhaber's formula

Faulhaber's formula concerns expressing the sum of the p-th powers of the first n positive integers$$\backslash sum\_^n\; k^p\; =\; 1^p\; +\; 2^p\; +\; 3^p\; +\; \backslash cdots\; +\; n^p$$as a (p + 1)th-degree polynomial function of n.

The first few examples are well known. For p = 0, we have$$\backslash sum\_^n\; k^0\; =\; \backslash sum\_^n\; 1\; =\; n\; .$$For p = 1, we have the triangular numbers$$\backslash sum\_^n\; k^1\; =\; \backslash sum\_^n\; k\; =\; \backslash frac\; =\; \backslash frac12(n^2\; +\; n)\; .$$For p = 2, we have the square pyramidal numbers$$\backslash sum\_^n\; k^2\; =\; \backslash frac\; =\; \backslash frac13(n^3\; +\; \backslash tfrac32\; n^2\; +\; \backslash tfrac12n)\; .$$

The coefficients of Faulhaber's formula in its general form involve the Bernoulli numbers B_j. The Bernoulli numbers beginwhere here we use the convention that $B\_1\; =\; +\backslash frac12$. The Bernoulli numbers have various definitions (see Bernoulli number#Definitions), such as that they are the coefficients of the exponential generating function$$\backslash frac\; =\; \backslash frac\; \backslash left(\backslash operatorname\; \backslash frac\; +1\; \backslash right)\; =\; \backslash sum\_^\backslash infty\; B\_k\; \backslash frac\; .$$

Then Faulhaber's formula is that$$\backslash sum\_^n\; k^\; =\; \backslash frac\; \backslash sum\_^p\; \backslash binom\; B\_k\; n^\; .$$Here, the B_j are the Bernoulli numbers as above, and$$\backslash binom\; =\; \backslash frac\; =\; \backslash frac$$is the binomial coefficient "p + 1 choose k".

Examples

So, for example, one has for,

The first seven examples of Faulhaber's formula are

History

Ancient period

The history of the problem begins in antiquity and coincides with that of some of its special cases. The case

p = 1 coincides with that of the calculation of the arithmetic series, the sum of the first n values of an arithmetic progression. This problem is quite simple but the case already known by the Pythagorean school for its connection with triangular numbers is historically interesting:1+2+...+ n = (1)/(2) n ^{2+ (1)/(2)} n, polynomial S ^{1 (n)} _{1 , 1} calculating the sum of the first n natural numbers.For m> 1 , the first cases encountered in the history of mathematics are:1+3+...+2 n -1= n ^{2,} polynomial S ^{1 (n)} _{1 , 2} calculating the sum of the first n successive odds forming a square. A property probably well known by the Pythagoreans themselves who, in constructing their figured numbers, had to add each time a gnomon consisting of an odd number of points to obtain the next perfect square.1 ^{2+2} ^{2+\ldots+n} ^{2= (1)/(3)} n ^{3+ (1)/(2)} n ^{2+ (1)/(6)} n, polynomial S ^{2 (n)} _{1 , 1} calculating the sum of the squares of the successive integers. Property that is demonstrated in Spirals, a work of Archimedes.1 ^{3+2} ^{3+\ldots+n} ^{3= (1)/(4)} n ^{4+ (1)/(2)} n ^{3+ (1)/(4)} n ^{2,} polynomial S ^{3 (n)} _{1 , 1} calculating the sum of the cubes of the successive integers. Corollary of a theorem of Nicomachus of Gerasa.^[1] L'insieme S ^{m(n)} _{1 , 1} of the cases, to which the two preceding polynomials belong, constitutes the classical problem of powers of successive integers.

Middle period

Over time, many other mathematicians became interested in the problem and made various contributions to its solution. These include Aryabhata, Al-Karaji, Ibn al-Haytham, Thomas Harriot, Johann Faulhaber, Pierre de Fermat and Blaise Pascal who recursively solved the problem of the sum of powers of successive integers by considering an identity that allowed to obtain a polynomial of degree

m + 1 already knowing the previous ones.^[1]

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.^[2]

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:

\sum ^{n} _{k =1} k ^{p} = (n ^{p+1})/(+ p +1) (1)/(2) n ^{p+{1} \over p +1 } \sum ^{p} _{j =2} {p +1 \choose j} B _{j} n ^{p+1-j} .

Introducing also the first two Bernoulli numbers (which Bernoulli did not), the previous formula becomes$$\backslash sum\_^n\; k^p\; =\; \backslash sum\_^p\; B\_j\; n^,$$using the Bernoulli number of the second kind for which $B\_1=\backslash frac$, or$$\backslash sum\_^n\; k^p\; =\; \backslash sum\_^p\; (-1)^j\; B\_j^-\; n^,$$using the Bernoulli number of the first kind for which $B\_1^-\; =-\backslash frac.$

A rigorous proof of these formulas and Faulhaber's assertion that such formulas would exist for all odd powers took until, two centuries later. Jacobi benefited from the progress of mathematical analysis using the development in infinite series of an exponential function generating Bernoulli numbers.

Modern period

In 1982 A.W.F. Edwards publishes an article ^[3] in which he shows that Pascal's identity can be expressed by means of triangular matrices containing the Pascal's triangle deprived of 'last element of each line:

\begin{pmatrix} n\\ n ^{2\\ n} ^{3\\ n} ^{4\\ n} ^{5\\ \end{pmatrix}=\begin{pmatrix} 1} & 0 & 0 & 0 & 0 \\ 1 & 2 & 0 & 0 & 0 \\ 1 & 3 & 3 & 0 & 0 \\ 1 & 4 & 6 & 4 & 0 \\ 1 & 5 & 10 & 10 & 5 \end{pmatrix}\begin{pmatrix} n\\ \sum ^{n -1} _{k =0} k ^{n -1} _{k =0} k ^{n -1} _{k =0} k ^{n -1} _{k =0} k ^{4\\ \end{pmatrix}} ^{[4] [5]}

The example is limited by the choice of a fifth order matrix but is easily extendable to higher orders. The equation can be written as:

\vec{N} = A\vec{S} and multiplying the two sides of the equation to the left by A ^{-1} , inverse of the matrix A, we obtain A ^{-1} \vec{N} = \vec{S} which allows to arrive directly at the polynomial coefficients without directly using the Bernoulli numbers. Other authors after Edwards dealing with various aspects of the power sum problem take the matrix path ^[6] and studying aspects of the problem in their articles useful tools such as the Vandermonde vector.^[7] Other researchers continue to explore through the traditional analytic route ^[8] and generalize the problem of the sum of successive integers to any geometric progression^{[9] [10]}

Proof with exponential generating function

Let $$S\_(n)=\backslash sum\_^\; k^p,$$denote the sum under consideration for integer

p\ge 0 .

Define the following exponential generating function with (initially) indeterminate

z $$G(z,n)=\backslash sum\_^\; S\_(n)\; \backslash fracz^p.$$We find This is an entire function in z so that z can be taken to be any complex number.B _{j(x)} $$\backslash frac=\backslash sum\_^\; B\_j(x)\; \backslash frac,$$where B _{j=B} _{j(0)} denotes the Bernoulli number with the convention B _{1} =- (1)/(2) . This may be converted to a generating function with the convention B ^{+} _{1= (1)/(2)} by the addition of j to the coefficient of x ^{j-1} in each B _{j(x)} , see Bernoulli_polynomials#Explicit_formula for example. B _{0} does not need to be changed.so that

$$\backslash sum\_^\; B^+\_j(x)\; \backslash frac\; -\; \backslash sum\_^\; B^+\_j(0)\; \backslash frac=\; \backslash frac\; -\; \backslash frac=\; zG(z,n)$$It follows that$$S\_p(n)=\backslash frac$$for all

p .

Faulhaber polynomials

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

Write$$a\; =\; \backslash sum\_^n\; k\; =\; \backslash frac\; .$$Faulhaber observed that if p is odd then $\backslash sum\_^n\; k^p$ is a polynomial function of a.

For p = 1, it is clear that$$\backslash sum\_^n\; k^1\; =\; \backslash sum\_^n\; k\; =\; \backslash frac\; =\; a.$$For p = 3, the result that$$\backslash sum\_^n\; k^3\; =\; \backslash frac\; =\; a^2$$is known as Nicomachus's theorem.

Further, we have(see,,,,).

More generally, $$\backslash sum\_^n\; k^\; =\; \backslash frac\; \backslash sum\_^m\; \backslash binom(2-2^)~\; B\_\; ~\backslash left[(8a+1)^\{m+1-q\}-1\backslash right].$$

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 .

Inversely, writing for simplicity

s _{j:} = \sum ^{n} _{k =1} k ^{j} , we have and generally$$2^\; a^m\; =\; \backslash sum\_\; \backslash binom\; s\_.$$

Faulhaber also knew that if a sum for an odd power is given by$$\backslash sum\_^n\; k^\; =\; c\_1\; a^2\; +\; c\_2\; a^3\; +\; \backslash cdots\; +\; c\_m\; a^$$then the sum for the even power just below is given by$$\backslash sum\_^n\; k^\; =\; \backslash frac(2\; c\_1\; a\; +\; 3\; c\_2\; a^2+\backslash cdots\; +\; (m+1)\; c\_m\; a^m).$$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 + 1/2 and n + 1.

Expressing products of power sums as linear combinations of power sums

Products of two (and thus by iteration, several) power sums

s _{j _{r}} : = \sum ^{n} _{k =1} k ^{j _{r}} can be written as linear combinations of power sums with either all degrees even or all degrees odd, depending on the total degree of the product as a polynomial in n , e.g. 30 s _{2s} _{4=-s} _{3+15s} _{5+16s} _{7} . Note that the sums of coefficients must be equal on both sides, as can be seen by putting n =1 , which makes all the s _{j} equal to 1. Some general formulae include:Note that in the second formula, for even m the term corresponding to j = \dfrac m 2 is different from the other terms in the sum, while for odd m , this additional term vanishes because of B _{m=0} .

Matrix form

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$$\backslash begin\backslash sum\; k^0\; \backslash \backslash \backslash sum\; k^1\; \backslash \backslash \backslash sum\; k^2\; \backslash \backslash \backslash sum\; k^3\; \backslash \backslash \backslash sum\; k^4\; \backslash \backslash \backslash sum\; k^5\; \backslash \backslash \backslash sum\; k^6\; \backslash end\; =G\_7\backslash beginn\; \backslash \backslash n^2\; \backslash \backslash n^3\; \backslash \backslash n^4\; \backslash \backslash n^5\; \backslash \backslash n^6\; \backslash \backslash n^7\; \backslash end,$$where

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

A _{7} be the matrix obtained from \overline{A} _{7} by changing the signs of the entries in odd diagonals, that is by replacing a _{i,j} by ( -1 ) ^{i+j} a _{i,j} , let \overline{G} _{7} be the matrix obtained from G _{7} with a similar transformation, thenandAlso$$\backslash begin\backslash sum\_^\; k^0\; \backslash \backslash \backslash sum\_^\; k^1\; \backslash \backslash \backslash sum\_^\; k^2\; \backslash \backslash \backslash sum\_^\; k^3\; \backslash \backslash \backslash sum\_^\; k^4\; \backslash \backslash \backslash sum\_^\; k^5\; \backslash \backslash \backslash sum\_^\; k^6\; \backslash \backslash \backslash end\; =\backslash overline\_7\backslash beginn\; \backslash \backslash n^2\; \backslash \backslash n^3\; \backslash \backslash n^4\; \backslash \backslash n^5\; \backslash \backslash n^6\; \backslash \backslash n^7\; \backslash \backslash \backslash end$$This is because it is evident that$\backslash sum\_^k^m-\backslash sum\_^k^m=n^m$ and that therefore polynomials of degree m + 1 of the form $\backslash fracn^+\backslash fracn^m+\backslash cdots$ subtracted the monomial difference n ^{m} they become $\backslash fracn^-\backslash fracn^m+\backslash cdots$.

This is true for every order, that is, for each positive integer, one has

G ^{-1} _{m} = \overline{A} _{m} and \overline{G} ^{-1} _{m} = A _{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 inverting the matrix easily obtained from the triangle of Pascal.^{[11] [12]}

Variations

• Replacing

k with p - k , we find the alternative expression: $$\backslash sum\_^n\; k^p=\; \backslash sum\_^p\; \backslash frac1\; B\_\; n^.$$

• Subtracting

n ^{p} from both sides of the original formula and incrementing n by 1 , we get

where

( -1 ) ^{kB} _{k} = B ^{-} _{k} can be interpreted as "negative" Bernoulli numbers with B ^{-} _{1=- \tfrac 12} .

• We may also expand

G(z,n) in terms of the Bernoulli polynomials to find which implies $$\backslash sum\_^n\; k^p\; =\; \backslash frac\; \backslash left(B\_(n+1)-(-1)^B\_\backslash right)\; =\; \backslash frac\backslash left(B\_(n+1)-B\_(1)\; \backslash right).$$ Since B _{n} = 0 whenever n > 1 is odd, the factor ( -1 ) ^{p+1} may be removed when p > 0 .

• It can also be expressed in terms of Stirling numbers of the second kind and falling factorials as^[13] $$$$

\sum_^n k^p = \sum_^p \left\\frac, $$\backslash sum\_^n\; k^p\; =\; \backslash sum\_^\; \backslash left\backslash \backslash frac.$$ 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.

Interpreting the Stirling numbers of the second kind,

\left\{{p +1 \atop k}\right\} , as the number of set partitions of \lbrack p +1 \rbrack into k parts, the identity has a direct combinatorial proof since both sides count the number of functions f:\lbrack p +1 \rbrack \to \lbrack n\rbrack with f( 1 ) maximal. The index of summation on the left hand side represents k = f( 1 ) , while the index on the right hand side is represents the number of elements in the image of f.

• There is also a similar (but somehow simpler) expression: using the idea of telescoping and the binomial theorem, one gets Pascal's identity:^[14]

This in particular yields the examples below – e.g., take to get the first example. In a similar fashion we also find$$\backslash beginn^\; =\; \backslash sum\_^n\; \backslash left(m^\; -\; (m-1)^\backslash right)\; =\; \backslash sum\_^k\; (-1)^\backslash binom\; (1^p+2^p+\; \backslash dots\; +\; n^p).\backslash end$$

A _{n(x)} is\sum ^{inf ty} _{n =1} n ^{k} x ^{n = (x)/(( 1- x) ^{k+1})} A _{k(x)} .

• Faulhaber's formula was generalized by Guo and Zeng to a -analog.^[15]

Relationship to Riemann zeta function

Using

B _{k=-k\zeta(1-k)} , one can write$$\backslash sum\backslash limits\_^n\; k^p\; =\; \backslash frac\; -\; \backslash sum\backslash limits\_^\backslash zeta(-j)n^.$$

If we consider the generating function

G(z,n) in the large n limit for \Re (z)< 0 , then we find$$\backslash lim\_G(z,n)\; =\; \backslash frac=\backslash sum\_^\; (-1)^B\_j\; \backslash frac$$Heuristically, this suggests that $$\backslash sum\_^\; k^p=\backslash frac.$$This result agrees with the value of the Riemann zeta function $\backslash zeta(s)=\backslash sum\_^\backslash frac$ for negative integers s =- p< 0 on appropriately analytically continuing \zeta(s) .

Faulhaber's formula can be written in terms of the Hurwitz zeta function:

$$\backslash sum\backslash limits\_^n\; k^p\; =\; \backslash zeta(-p)\; -\; \backslash zeta(-p,\; n+1)$$

Umbral form

In the umbral calculus, one treats the Bernoulli numbers $B^0\; =\; 1$, $B^1\; =\; \backslash frac$, $B^2\; =\; \backslash frac$, ... as if the index j in $B^j$ 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$$\backslash sum\_^n\; k^p\; =\; \backslash frac\; \backslash big((B+n)^\; -\; B^\; \backslash big)\; .$$Here, the expression on the right must be understood by expanding out to get terms $B^j$ 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.^[16]

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 $T(b^j)\; =\; B\_j.$ Then one can say

A general formula

The series

1 ^{m} + 2 ^{m} + 3 ^{m} + ... + n ^{m} as a function of m is often abbreviated as S _{m} . Beardon has published formulas for powers of S _{m} , including a 1996 paper^[17] which demonstrated that integer powers of S _{1} can be written as a linear sum of terms in the sequence S _{3,} S _{5,} S _{7, ...} :S ^{N} _{1} = (1)/(2 ^{N}) \sum ^{N} _{r =0} {N \choose r}S _{N+r} \left( 1- ( -1 ) ^{N-r} \right)

The first few resulting identities are then

S ^{2} _{1} = S _{3} S ^{3} _{1} = (1)/(4) S _{3} + (3)/(4) S _{5} S ^{4} _{1} = (1)/(2) S _{5} + (1)/(2) S _{7} .

Although other specific cases of

S ^{N} _{m}  - including S ^{2} _{2} = (1)/(3) S _{4} + (2)/(3) S _{5} and S ^{3} _{2} = (1)/(12) S _{4} + (7)/(12) S _{6+} (1)/(3) S _{8}  - are known, no general formula for S ^{N} _{m} for positive integers m and N has yet been reported. A 2019 paper by Derby^[18] proved that: S ^{N} _{m} = \sum ^{N} _{k =1} ( -1 ) ^{k-1} {N \choose k}\sum ^{n} _{r =1} r ^{mk} S ^{N - k} _{m} (r) .

This can be calculated in matrix form, as described above. The

m = 1 case replicates Beardon's formula for S ^{N} _{1} and confirms the above-stated results for m = 2 and N = 2 or 3 . Results for higher powers include:S ^{4} _{2} = (1)/(54) S _{5} + (5)/(18) S _{7} + (5)/(9) S _{9} + (4)/(27) S _{11} S ^{3} _{6} = (1)/(588) S _{8} - (1)/(42) S _{10} + (13)/(84) S _{12} - (47)/(98) S _{14} + (17)/(28) S _{16} + (19)/(28) S _{18} + (3)/(49) S _{20} .

External links

• News: Jacobi . Carl . 1834 . De usu legitimo formulae summatoriae Maclaurinianae . . 12 . 263–72 . 10.1515/crll.1834.12.263 .
  • Book: Johann Faulhaber. Academia Algebrae - Darinnen die miraculosische Inventiones zu den höchsten Cossen weiters continuirt und profitiert werden . 1631. A very rare book, but Knuth has placed a photocopy in the Stanford library, call number QA154.8 F3 1631a f MATH.
• Beardon. A. F.. Sums of Powers of Integers. American Mathematical Monthly. 103. 3. 1996. 201–213. 2011-10-23. 10.1080/00029890.1996.12004725. (Winner of a Lester R. Ford Award)
• News: Schumacher . Raphael . 2016 . An Extended Version of Faulhaber's Formula . Journal of Integer Sequences . 19 . 16.4.2.
• News: Orosi . Greg . 2018 . A Simple Derivation Of Faulhaber's Formula . Applied Mathematics E-Notes . 18 . 124–126.
• A visual proof for the sum of squares and cubes.

Notes and References

1. News: Janet. Beery. Sum of powers of positive integers. 2009. MMA Mathematical Association of America . 10.4169/loci003284. 1 November 2024.
2. Donald E. Knuth . Johann Faulhaber and sums of powers . Mathematics of Computation . 1993 . 61 . 277 - 294 . math.CA/9207222. 10.2307/2152953. 203. 2152953. Donald E. Knuth . 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.
3. Anthony William Fairbank. Edwards . Sums of powers of integers: A little of the History . The Mathematical Gazette. 66. 435 . 1982. 22–28 . 10.2307/3617302. 3617302 . 125682077 .
4. The first element of the vector of the sums is n and not \sum ^{n -1} _{k =0} k ^{0} because of the first addend, the indeterminate form 0 ^{0} , which should otherwise be assigned a value of 1
5. Book: Edwards, A.W.F.. Pascal's Arithmetical Triangle: The Story of a Mathematical Idea. 84. Charles Griffin & C.. 1987. 0-8018-6946-3.
6. News: Dan. Kalman . Sums of Powers by matrix method . Semantic scholar . 1988. 2656552 .
7. News: Gottfried. Helmes . Accessing Bernoulli-Numbers by Matrix-Operations . Uni-Kassel.de . 2006.
8. Web site: F.T. Howard. Sums of powers of integers via generating functions. 1994. 10.1.1.376.4044.
9. Wolfdieter. Lang . On Sums of Powers of Arithmetic Progressions, and Generalized Stirling, Eulerian and Bernoulli numbers. 2017 . math.NT . 1707.04451 .
10. Do. Tan Si. Obtaining Easily Sums of Powers on Arithmetic Progressions and Properties of Bernoulli Polynomials by Operator Calculus. Canadian Center of Science and Education. Applied Physics Research. 1916-9639. 9. 2017.
11. .
12. .
13. [Concrete Mathematics]
14. Kieren MacMillan, Jonathan Sondow. Proofs of power sum and binomial coefficient congruences via Pascal's identity . . 2011 . 118 . 6 . 549 - 551 . 10.4169/amer.math.monthly.118.06.549. 1011.0076. 207521003 .
15. 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 . 10.37236/1876 . 2005math......1441G . math/0501441 . 10467873 .
16. Book: . The Book of Numbers . Springer . 1996 . 0-387-97993-X . 107 .
17. The American Mathematical Monthly. 103. 1996. 3. Sums of Powers of Integers. A. F.. Beardon. 201–213. 10.1080/00029890.1996.12004725.
18. Nigel M.. Derby. The continued search for sums of powers. 2019. The Mathematical Gazette. 103. 556. 94–100. 10.1017/mag.2019.11.