So that I don't forget how to do these, here are the few methods of analytical continuation I have discovered and want to prove.

1: Eta/Zeta method-- Replace all sum(n^k) or sum( (-1)^k n^k) with the eta/zeta version of those sums. This essentially analytically continues the bases of sum(n^k) into their correct rational functions. This will fail if the divergence of sum(n^k) is somehow crucial to how the sum plays out. This should always work if the power series for f(n) has a radius of convergence which converges very quickly.

2:Li(x) method-- Replace sum(x^n n^k) with LI. This is very similar to the previous method, except that it replaces different rational functions with slightly different bases

3: Smooth method-- multiply f(n) by any sufficiently fast decaying and smooth cutoff functions will analytically continue it if it doesn't have a natural boundary

4: 'Natural' laurent series-- If one can find a natural function for f(n) which extends to f(-n), this function often extends the function everywhere, even past singluarities. This is the most promising method to extend past singularities. One potential pattern is that any composition or multiplication of trig functions seems to get something valid, so maybe any series which can expressed as a finite composition of those basis functions can be continued, even if they have a natural boundary