Author of this book i am reading claims that the nth factor of $\frac {1\cdot 2\cdot 3\cdots( m-1) } {( z+1)( z+2) \cdots( z+m-1) }m^{s}$ is $\frac {n} {z+n}( \frac {n+1} {n}) ^{s}$ shouldn't it be just $\frac {n} {z+n}( n+1) ^{s} ?$
Edit: The author claims the product would look like $\prod _{n=1}^{n=m-1}\frac {n} {z+n}\left( \frac {n+1} {n}\right) ^{s}.$ I think (most likely incorrectly) that it should be written as $\prod _{n=1}^{n=m-1}\frac {n} {z+n}\left({n+1} \right) ^{s}$
i think i can see my mistake so i am going to close the post.