Is there any property of Gamma Function that simplifies this expression?

Question

The expression I'd like to simplify is $\frac{\Gamma{[1+q\cdot x]}}{\Gamma{[1+x]^{q}}}$, If somebody could give me a hint I will appreciate it.

Thanks in advance

Answer 1

I don't know how helpful this will be, but you could use $\Gamma\left( 1+z \right)=z\Gamma\left( z \right),\,B\left( a,\,b\right) =\frac{\Gamma\left( a \right)\Gamma\left( b \right)}{\Gamma\left( a+b \right)}$ to write $$\frac{\Gamma\left( 1+qx \right)}{\Gamma^q\left( 1+x \right)}=\frac{qx^{1-q}}{\prod_{j=1}^{q-1}B\left( jx,\,x\right)}.$$

Answer 2

Do you know a simplification for the special case $$\frac{(3n)!}{n!^3}\quad?$$ Actually, I think that is probably the simplest way to write it.

You could say: in $(a+b+c)^{3n}$, take the coefficient of the term $a^nb^nc^n$. You can try to do something similar in your general case.