Is it possible to simplify this expression?

Γ(110)Γ(215) Γ(715)

Is there a systematic way to check ratios of Gamma-functions like this for simplification possibility?

**Answer**

Amazingly, this can be greatly simplified. I’ll state the result first:

Γ(110)Γ(215)Γ(715)=√5+131/1026/5√π

The result follows first from a version of Gauss’s multiplication formula:

Γ(3z)=12π33z−1/2Γ(z)Γ(z+13)Γ(z+23)

or, with z=2/15:

Γ(215)Γ(715)=2π31/10Γ(25)Γ(45)

Now use the duplication formula

Γ(2z)=1√π22z−1Γ(z)Γ(z+12)

or, with z=2/5:

Γ(25)Γ(45)=√π21/5Γ(910)

Putting this all together, we get

Γ(110)Γ(215)Γ(715)=Γ(110)Γ(910)√π326/531/10

And now, we may use the reflection formula:

Γ(z)Γ(1−z)=πsinπz

With z=1/10, and noting that

sin(π10)=√5−14=1√5+1

the stated result follows. This has been verified numerically in Wolfram|Alpha.

**Attribution***Source : Link , Question Author : X.C. , Answer Author : Ron Gordon*