Gauss's Hypergeometric Theorem/Corollary 1

Corollary to Gauss's Hypergeometric Theorem
Let $\map \Re {1 - a} > 0$.

Let $c \notin \Z_{\le 0}$ and $c \ne 1$.

Then:


 * $\ds \sum_{k \mathop = 0}^\infty \dfrac {a^{\overline k} } {\paren {c - 1 + k} k!} = \dfrac {\map \Gamma {c - 1} \map \Gamma {1 - a} } {\map \Gamma {c - a} }$

Proof
Set $b = c - 1$ in Gauss's Hypergeometric Theorem

Before substitution:

After substitution: