Digital Library of Mathematical Functions
The Gamma Function -- Infinite Products

### Infinite Products

Γ ( z ) = lim k → ∞ k ! k z z ( z + 1 ) ⋯ ( z + k )
z ≠ 0 , - 1 , - 2 , … ,
1 Γ ( z ) = z ⅇ γ z ∏ k = 1 ∞ ( 1 + z k ) ⅇ - z k
| Γ ( x ) Γ ( x + ⅈ y ) | 2 = ∏ k = 0 ∞ ( 1 + y 2 ( x + k ) 2 ) ,
x ≠ 0 , - 1 , … .
∑ k = 1 m a k = ∑ k = 1 m b k

then

∏ k = 0 ∞ ( a 1 + k ) ( a 2 + k ) ⋯ ( a m + k ) ( b 1 + k ) ( b 2 + k ) ⋯ ( b m + k ) = Γ ( b 1 ) Γ ( b 2 ) ⋯ Γ ( b m ) Γ ( a 1 ) Γ ( a 2 ) ⋯ Γ ( a m )

provided that none of the b k is zero or a negative integer.