I have trouble with the following problem.
Let $f=f(p)$, $p>1$, $0
I have trouble with the following problem.
Let $f=f(p)$, $p>1$, $0
For positive half-integers, the gamma function is given by $ \Gamma(\frac{1}{2} + n) = \frac{(2n)!}{4^nn!}\sqrt{\pi}, $ and so the ratio in question is $ \frac{\Gamma(p+1)}{\Gamma(p+\frac{3}{2})} = \frac{4^{p+1}p!(p+1)!}{(2p+2)!\sqrt{\pi}} $ for integral $p$. As $p\rightarrow\infty$, this is asymptotic to $p^{-1/2}$, so the limit in the problem becomes $ \lim_{p\rightarrow\infty}p\left(1-\sqrt{f(p)}\right). $ To make this take on a particular finite value $\alpha$, the expression $1-\sqrt{f(p)}$ must be asymptotic to $\alpha/p$, hence $\sqrt{f(p)}$ must be asymptotic to $1-\alpha/p$, hence $f(p)$ must be asymptotic to $1-2\alpha/p$. Setting $\alpha=3/20$, we have $ f(p) = 1-\frac{3}{10p} $ as one function that satisfies the problem.