Many special functions including the gamma function have a duplication formula of some sorts. In the case of the gamma function it reads:

Gamma(2z) = Gamma(z) Gamma(z+1/2) 2^2z-1/Gamma(1/2)

On the other hand, there is no algebraic relation between Gamma(2z) and Gamma(z) by themselves, meaning that is there is no nonzero polynomial f(x,y) such that f(Gamma(2z),Gamma(z))=0 for all complex z. I can prove this by chasing poles and their order.

However, I'd be interested in a (simple) argument which shows that the following similar statement is true (which I believe it is):

There is no (nonzero) polynomial f(x,y) such that f((2n)!, n!)=0 for all integers n≥0.

Any ideas? Thank you!