Gamma Difference Equation
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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
By Euler's form of the Gamma function:
\(\ds \frac {\map \Gamma {z + 1} } {\map \Gamma z}\) | \(=\) | \(\ds \paren {\frac 1 {z + 1} \lim_{m \mathop \to \infty} \prod_{n \mathop = 1}^m \frac {\paren {1 + \frac 1 n}^{z + 1} } {1 + \frac {z + 1} n} } \div \paren {\frac 1 z \lim_{m \mathop \to \infty} \prod_{n \mathop = 1}^m \frac {\paren {1 + \frac 1 n}^z} {1 + \frac z n} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac z {z + 1} \lim_{m \mathop \to \infty} \prod_{n \mathop = 1}^m \paren {\frac {\paren {1 + \frac 1 n}^{z + 1} \paren {1 + \frac z n} } {\paren {1 + \frac 1 n}^z \paren {1 + \frac {z + 1} n} } }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac z {z + 1} \lim_{m \mathop \to \infty} \prod_{n \mathop = 1}^m \paren {\frac {\paren {1 + \frac 1 n} \paren {z + n} } {z + n + 1} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds z \lim_{m \mathop \to \infty} \frac {m + 1} {z + m + 1} = z\) |
$\blacksquare$
Sources
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- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 16$: The Gamma Function: Recursion Formula: $16.2$
- 1983: François Le Lionnais and Jean Brette: Les Nombres Remarquables ... (previous) ... (next): $0,88560 31944 \ldots$
- 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.5$: Permutations and Factorials: Exercise $10$
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): gamma function
- 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $11.14 \ \text{(iii)}$
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): gamma function
- 2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 25$: The Gamma Function: Recursion Formula: $25.2.$