Digital Library of Mathematical Functions
About the Project
NIST
5 Gamma FunctionProperties

§5.8 Infinite Products

5.8.1 Γ(z)=limkk!kzz(z+1)(z+k),
z0,-1,-2,,
5.8.2 1Γ(z)=zγzk=1(1+zk)-z/k,
5.8.3 |Γ(x)Γ(x+y)|2=k=0(1+y2(x+k)2),
x0,-1,.

If

5.8.4 k=1mak=k=1mbk,

then

5.8.5 k=0(a1+k)(a2+k)(am+k)(b1+k)(b2+k)(bm+k)=Γ(b1)Γ(b2)Γ(bm)Γ(a1)Γ(a2)Γ(am),

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