The series below converges to a familiar analytic function in some open half plane. Which half plane and which function?
$$\sum\limits_{n=0}^\infty {\frac{z(z+1)\cdots(z+(n-1))}{n!}}=1+z+\frac12z(z+1)+\frac16z(z+1)(z+2)+\dots$$
In class we only cover that the power series converge in a circle of complex plane, so I do not have any clue about the series like this, any helpful hint or advice is welcome! Thanks in advance
Hint:
note that, by using the notation of the Rising and Falling Factorials we can rewrite your sum as $$ \eqalign{ & F(z) = \sum\limits_{0\, \le \,n} {{{z\left( {z + 1} \right) \cdots \left( {z + n - 1} \right)} \over {n!}}} = \cr & = \sum\limits_{0\, \le \,n} {{{z^{\,\overline {\,n\,} } } \over {n!}}} = \sum\limits_{0\, \le \,n} {{{\left( {z + n - 1} \right)^{\,\underline {\,n\,} } } \over {n!}}} = \cr & = \sum\limits_{0\, \le \,n} {\left( \matrix{ n + z - 1 \cr n \cr} \right)} = \sum\limits_{0\, \le \,n} {\left( { - 1} \right)^{\,\,n} \left( \matrix{ - z \cr n \cr} \right)} \quad \to \cr & \to \quad \left( {1 - 1} \right)^{\, - \,z} \cr} $$