Faulhaber's Formula - Alternate Expression

Alternate Expression

Due to the relationship between the Riemann zeta function and Bernoulli numbers, and the observations below, an arguably more natural expression in terms of B1=1/2 is

The older expression given earlier follows from this one as a result of the (somewhat coincidental) fact that when the sum Sp(n) is extended to negative integers using the property Sp(n−1)=Sp(n)−np, we have

\begin{align} S_p(-n) & = S_p(-1)-\sum_{k=1}^{n-1}(-k)^p=S_p(-1)-\left(\sum_{k=1}^n(-k)^p-(-n)^p\right) \\ & = S_p(-1)-(-1)^p\left(\sum_{k=1}^nk^p-n^p\right). \end{align}

Since Sp(−1)=Sp(0)−0p, it is 0 for p>0, but is indeterminate when p=0. S0(n)=n, however, so the convention is that Sp(−1) is the negated Kronecker delta, giving

The subtraction of np (when it is not cancelled by the delta for p=0) changes the sign of B1 (it is the coefficient of np), giving

in terms of B1=−1/2, which is essentially the same as the conventional expression given earlier.

The use of this expression at n=−1 also gives

for a recurrence relation giving the B1=−1/2 sequence.

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