# Is it possible to simplify Γ(110)Γ(215) Γ(715)\frac{\Gamma\left(\frac{1}{10}\right)}{\Gamma\left(\frac{2}{15}\right)\ \Gamma\left(\frac{7}{15}\right)}?

Is it possible to simplify this expression?

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:

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

or, with $z=2/15$:

Now use the duplication formula

or, with $z=2/5$:

Putting this all together, we get

And now, we may use the reflection formula:

With $z=1/10$, and noting that

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

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