Bernoulli Number - Sum of Powers

Sum of Powers

Bernoulli numbers feature prominently in the closed form expression of the sum of the m-th powers of the first n positive integers. For m, n ≥ 0 define

This expression can always be rewritten as a polynomial in n of degree m + 1. The coefficients of these polynomials are related to the Bernoulli numbers by Bernoulli's formula:

where the convention B₁ = +1/2 is used. ( denotes the binomial coefficient, m+1 choose k.)

For example, taking m to be 1 gives the triangular numbers 0, 1, 3, 6, ... (sequence A000217 in OEIS).

Taking m to be 2 gives the square pyramidal numbers 0, 1, 5, 14, ... (sequence A000330 in OEIS).

Some authors use the convention B₁ = −1/2 and state Bernoulli's formula in this way:

.

Bernoulli's formula is sometimes called Faulhaber's formula after Johann Faulhaber who also found remarkable ways to calculate sum of powers.

Faulhaber's formula was generalized by V. Guo and J. Zeng to a q-analog (Guo & Zeng 2005).