Digital Library of Mathematical Functions
The Gamma Function -- Function Relations

### Functional Relations

###### Contents

- Recurrence
- Reflection
- Multiplication
- Bohr-Mollerup Theorem

#### Recurrence

Γ ( z + 1 ) = z Γ ( z )
ψ ( z + 1 ) = ψ ( z ) + 1 z

#### Reflection

Γ ( z ) Γ ( 1 - z ) = π sin ( π z )
z ≠ 0 , ŷ 1 , … ,
ψ ( z ) - ψ ( 1 - z ) = - π tan ( π z )
z ≠ 0 , ŷ 1 , … .

#### Multiplication

2 z ≠ 0 , - 1 , - 2 , … ,
Γ ( 2 z ) = π - 1 2 2 2 z - 1 Γ ( z ) Γ ( z + 1 2 )
3 z ≠ 0 , - 1 , - 2 , … ,
Γ ( 3 z ) = ( 2 π ) - 1 3 3 z - ( 1 2 ) Γ ( z ) Γ ( z + 1 3 ) Γ ( z + 2 3 )
n z ≠ 0 , - 1 , - 2 , … ,
Γ ( n z ) = ( 2 π ) ( 1 - n ) 2 n n z - ( 1 2 ) ∏ k = 0 n - 1 Γ ( z + k n )
∏ k = 1 n - 1 Γ ( k n ) = ( 2 π ) ( n - 1 ) 2 n - 1 2
ψ ( 2 z ) = 1 2 ( ψ ( z ) + ψ ( z + 1 2 ) ) + ln 2
ψ ( n z ) = 1 n ∑ k = 0 n - 1 ψ ( z + k n ) + ln n

#### Bohr-Mollerup Theorem

If a positive function f ( x ) on ( 0 , ∞ ) satisfies f ( x + 1 ) = x f ( x ) , f ( 1 ) = 1 , and ln f ( x ) is convex, then f ( x ) = Γ ( x ) .