# Gamma Difference Equation

## Theorem

The Gamma function satisfies:

$\map \Gamma {z + 1} = z \, \map \Gamma z$

for any $z$ which is not a nonpositive integer.

## Proof 1

Let $z \in \C$, with $\map \Re z > 0$.

Then:

 $\ds \map \Gamma {z + 1}$ $=$ $\ds \int_0^\infty t^z e^{-t} \rd t$ $\ds$ $=$ $\ds \bigintlimits {-t^z e^{-t} } 0 \infty + z \int_0^\infty t^{z - 1} e^{-t} \rd t$ Integration by Parts $\ds$ $=$ $\ds z \int_0^\infty t^{z - 1} e^{-t} \rd t$ $\ds$ $=$ $\ds z \, \map \Gamma z$

If $z \in \C \setminus \set {0, -1, -2, \ldots}$ such that $\map \Re z \le 0$, then the statement holds by the definition of $\Gamma$ in this region.

$\blacksquare$

## Proof 2

 $\ds \frac {\map \Gamma {z + 1} } {\map \Gamma z}$ $=$ $\ds \paren {\frac 1 {z + 1} \lim_{m \mathop \to \infty} \prod_{n \mathop = 1}^m \frac {\paren {1 + \frac 1 n}^{z + 1} } {1 + \frac {z + 1} n} } \div \paren {\frac 1 z \lim_{m \mathop \to \infty} \prod_{n \mathop = 1}^m \frac {\paren {1 + \frac 1 n}^z} {1 + \frac z n} }$ $\ds$ $=$ $\ds \frac z {z + 1} \lim_{m \mathop \to \infty} \prod_{n \mathop = 1}^m \paren {\frac {\paren {1 + \frac 1 n}^{z + 1} \paren {1 + \frac z n} } {\paren {1 + \frac 1 n}^z \paren {1 + \frac {z + 1} n} } }$ $\ds$ $=$ $\ds \frac z {z + 1} \lim_{m \mathop \to \infty} \prod_{n \mathop = 1}^m \paren {\frac {\paren {1 + \frac 1 n} \paren {z + n} } {z + n + 1} }$ $\ds$ $=$ $\ds z \lim_{m \mathop \to \infty} \frac {m + 1} {z + m + 1} = z$

$\blacksquare$