Simplifying Series of Gamma Function I am trying to simplify summation of gamma function 
$$\sum_{k=1}^\infty\frac{\Gamma(k+b)}{(k-1)!\Gamma(b)} = \sum_{k=1}^\infty\frac{\Gamma(k+b)}{\Gamma(k)\Gamma(b)} , b > 1$$
I need hint/direction here. What is the property that I should use to simplify the expression into a non-summation form?
Thanks!

\begin{align}
S &= \sum_{k=1}^{\infty} \frac{\Gamma(k+b)}{(k-1)!\Gamma(b)} = \sum_{k=1}^{\infty} \frac{\Gamma(k+b)}{\Gamma(k)\Gamma(b)} \\
&= \sum_{k=0}^{\infty} \frac{b \, \Gamma(k+b+1)}{k! \, \Gamma(b+1)} \\
&= b \, \sum_{k=0}^{\infty} \frac{(b+1)_{k}}{k!} \\
&= b \, (1 - 1)^{-b-1}.
\end{align}
If $b$ is not negative then this series is infinite. Now, if $b = -a - n$, for $n \geq 1$, then the series is zero.
