Why Mathematica does nothing with the expression Gamma[2 z]? [closed]

Question

Could someone tell me why Mathematica just returns the input with this expression:

Gamma[2z]

Gamma[2 z]

?

I expected this result:

$Γ(2z)=\frac{2^{2z-1}Γ(z)Γ(z+1/2)}{\sqrt\pi}$

Answer 1

You can use an undocumented internal function to for this purpose:

Simplify`GammaTuplicate[Gamma[2z], 2] //TeXForm

$\frac{2^{2 z-1} \Gamma (z) \Gamma \left(z+\frac{1}{2}\right)}{\sqrt{\pi }}$

Another example:

Simplify`GammaTuplicate[Gamma[3z], 3] //TeXForm

$\frac{3^{3 z-1/2} \Gamma (z) \Gamma \left(z+\frac{1}{3}\right) \Gamma \left(z+\frac{2}{3}\right)}{2 \pi }$

Answer 2

A top level way to look up the identity is through MathematicalFunctionData:

identity = MathematicalFunctionData[Gamma, "MultipliedArgumentFormulas", 
  "IncludedSubexpressions" -> {Gamma[2 _]}][[1]];

Activate[identity[z]]

Gamma[2 z] == (2^(-1 + 2 z) Gamma[z] Gamma[1/2 + z])/Sqrt[\[Pi]]

---

An obscure way of getting the identity is to take backward and forward Mellin transforms and to mix in some hackery that prevent simplifications along the way:

Block[{Simplify`SimplifyGamma = # &},
  MellinTransform[MeijerGReduce[InverseMellinTransform[Gamma[2 z], z, s], s], s, z]
]

 (2^(-1 + 2 z) Gamma[z] Gamma[1/2 + z])/Sqrt[\[Pi]]