Faulhaber's formula

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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.

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

B_0 = 1,\quad B_1 = -{1 \over 2},\quad B_2 = {1 \over 6}, \quad B_3 = 0,\quad B_4 = -{1 \over 30},\quad\dots

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

\sum_{k=1}^n k^p = {1 \over p+1} \sum_{j=0}^p {p+1 \choose j} B_j n^{p+1-j}\qquad \left(\mbox{with } B_1 = {1 \over 2} \mbox{ rather than }-{1 \over 2}\right)

(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 derivation of the Faulhaber's Formula is available in The Book of Numbers by John Horton Conway and Richard Guy.

Contents

[edit] The first several cases

1 + 2 + 3 + \cdots + n = {n(n+1) \over 2} = {n^2 + n \over 2}
1^2 + 2^2 + 3^2 + \cdots + n^2 = {n(n+1)(2n+1) \over 6} = {2n^3 + 3n^2 + n \over 6}
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}
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}

[edit] Another form

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

[edit] 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 φj is the jth Bernoulli polynomial.

[edit] Umbral form

In the classic 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

\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).

The derivation of the Faulhaber's Formula is available in The Book of Numbers by John Conway.

[edit] 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.\,

In particular

1^3 + 2^3 + 3^3 + \cdots + n^3 = a^2\,


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


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


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


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

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".

[edit] References and external links

  • Eric W. Weisstein, Faulhaber's formula at MathWorld.
  • "Darinnen die miraculosische Inventiones zu den höchsten Cossen weiters continuirt und profitiert werden", Academia Algebrae, Johann Faulhaber, Augpurg, bey Johann Ulrich Schöigs, 1631.