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5 Gamma FunctionProperties

§5.5 Functional Relations

Contents

§5.5(i) Recurrence

5.5.1 Γ(z+1)=zΓ(z),
5.5.2 ψ(z+1)=ψ(z)+1z.

§5.5(ii) Reflection

5.5.3 Γ(z)Γ(1-z)=π/sin(πz),
z0,±1,,
5.5.4 ψ(z)-ψ(1-z)=-π/tan(πz),
z0,±1,.

§5.5(iii) Multiplication

Duplication Formula

For 2z0,-1,-2,,

5.5.5 Γ(2z)=π-1/222z-1Γ(z)Γ(z+12).

Gauss’s Multiplication Formula

For nz0,-1,-2,,

5.5.6 Γ(nz)=(2π)(1-n)/2nnz-(1/2)k=0n-1Γ(z+kn).
5.5.7 k=1n-1Γ(kn)=(2π)(n-1)/2n-1/2.
5.5.8 ψ(2z)=12(ψ(z)+ψ(z+12))+ln2,
5.5.9 ψ(nz)=1nk=0n-1ψ(z+kn)+lnn.

See also Sándor and Tóth (1989).

§5.5(iv) Bohr–Mollerup Theorem

If a positive function f(x) on (0,) satisfies f(x+1)=xf(x), f(1)=1, and lnf(x) is convex (see §1.4(viii)), then f(x)=Γ(x).