Gamma function written as a limit of a factorial, exponential, and a rising factorial

Theorem

The following formula holds: $$\Gamma(z) = \displaystyle\lim_{n \rightarrow \infty} \dfrac{n! n^z}{z(z+1) \ldots (z+n)} = \displaystyle\lim_{n \rightarrow \infty} \dfrac{n^z}{z(1+z)(1+\frac{z}{2}) \ldots (1+\frac{z}{n})},$$ where $\Gamma$ denotes the gamma function.

References

• 1953: Arthur Erdélyi, Wilhelm Magnus, Fritz Oberhettinger and Francesco G. Tricomi: Higher Transcendental Functions Volume I ... (previous) ... (next): §1.1 (2)