Why Does This Polylogarithm Identity Have No Restrictions?

[rman144]

I found this equation last night on Wolfram:

http://functions.wolfram.com/ZetaFunctionsandPolylogarithms/PolyLog/06/03/0001/

How is it possible this equation has no restrictions given that the gamma function has poles at the negative integers?

Also, won't the zeta function portion run into problems at those integers as well when k passes v in the summation?

 

[HallsofIvy]

There are no restrictions because one side is not defined at exactly the points the other is not defined.