Why this always gives -1?

The following sum always gives -1 if k is undefined but different values when k is defined. Why?

Sum[DifferenceDelta[n^k, n], {n, 1, Infinity}, Regularization -> Dirichlet]

• Do not use the bugs tag until what you've observed has been confirmed by other users. – J. M.'s discontentment Apr 24 '17 at 13:48

Because:

DifferenceDelta[n^k, n]


gives:

n^k + (1 + n)^k


with k unspecified

Sum[  n^k,{n,1,Infinity}]


is (generically)

-Zeta[-k]


while

Sum[ (1+n)^k,{n,1,Infinity}]


is

-1 + Zeta[-k]


hence

Sum[DifferenceDelta[n^k, n], {n, 1, Infinity}]


gives -1.

The reason for this is that this is the answer for all Complex k for which the sum converges.