Share on Facebook Share on Twitter Email
Answers.com

Faulhaber's formula

 
Wikipedia: Faulhaber's formula
Home > Library > Miscellaneous > Wikipedia

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).

The derivation of the Faulhaber's Formula is available in The Book of Numbers by John Horton Conway and Richard Guy.[1]

Contents

Examples

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}

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.

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).

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". The polynomials in the right-hand sides are divisible by a 2 because for j > 1 odd the Bernoulli number Bj is 0.

History

Faulhaber's formula is also called Bernoulli's formula. That is more fair from historical perspective. 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.[2]

Faulhaber realized that for odd m, Sm(n) is not just a polynomial in n but a polynomial in the triangular number N = n(n + 1)/2. For example he found that:

1 + 2 + \cdots + n = N;
1^2 + 2^2 + \cdots + n^2 = N \left(2n+1\right) /3 ;
1^3 + 2^3 + \cdots + n^3 = N^2.

To call Bernoulli's formula Faulhaber's formula does injustice to Bernoulli and simultaneously hides the genius of Faulhaber as Faulhaber's formula is in fact more efficient than Bernoulli's formula. According to Knuth,[2] a rigorous proof of Faulhaber’s formula was first published by Carl Jacobi in 1834 (Jacobi 1834). Donald E. Knuth's in-depth study of Faulhaber's formula concludes:[2]

“Faulhaber never discovered the Bernoulli numbers; i.e., he never realized that a single sequence of constants B0, B1, B2, … would provide a uniform

\quad \sum n^m = \frac 1{m+1}\left( B_0n^{m+1}-\binom{m+1}1B_1n^m+\binom{m+1} 2B_2n^{m-1}-\cdots +(-1)^m\binom{m+1}mB_mn\right)

for all sums of powers. He never mentioned, for example, the fact that almost half of the coefficients turned out to be zero after he had converted his formulas for \sum n^m from polynomials in N to polynomials in n.”

References and external links

  1. ^ John H. Conway, Richard Guy (1998). The Book of Numbers. Springer. p. 107. ISBN 0-387-97993-X. 
  2. ^ a b c Donald E. Knuth (1993). "Johann Faulhaber and sums of powers". Math. Comp. 61 (203): 277–294. http://arxiv.org/abs/math.CA/9207222. 

This entry is from Wikipedia, the leading user-contributed encyclopedia. It may not have been reviewed by professional editors (see full disclaimer)

Search unanswered questions...

Enter a question here...
Search: All sources Community Q&A Reference topics

 
 

 

Copyrights:

Wikipedia. This article is licensed under the Creative Commons Attribution/Share-Alike License. It uses material from the Wikipedia article "Faulhaber's formula".  Read more