How could I approach a limit question of the Barnes G-function?

Question

The exact question I have is $$\lim_{n\to\infty} \frac{G(n+\frac{5}{4})G(n+\frac{7}{4})^{2}G(n+\frac{9}{4})}{G(n+2)^{2}G(n+\frac{3}{2})^{2}}$$

Intuitively it looks like it should tend to $1$ but I do not have experience with the Barnes G-function.

Answer 1

Use the known asymptotics $$ \ln G(1+z)\sim z^2\left(\frac{1}{2}\ln z-\frac{3}{4}\right)+\frac{1}{2}\ln (2\pi)z-\frac{1}{12}\ln z+\ldots $$ to deal with your limit $\lim_{n\to\infty}f(n)$. Consider $\ln f(n)$: $$ \ln f(n)=\ln G((n+1/4)+1)+2\ln G((n+3/4)+1)+\ln G((n+5/4)+1)-2\ln G((n+1)+1)-2\ln G((n+1/2)+1) $$ $$ \sim \frac{1}{8}\ln n\ , $$ from which one deduce that $f(n)\sim n^{1/8}\to \infty$.

Answer 2

In the same spirit as Pierpaolo Vivo's answer, using the given asymptotics with more terms, you can have a quite accurate approximation of $$f_n=\frac{G(n+\frac{5}{4})\,G(n+\frac{7}{4})^{2}\,G(n+\frac{9}{4})}{G(n+2)^{2}\,G(n+\frac{3}{2})^{2}}$$ even for small values of $n$.

Expanding $\log(f_n)$ and using later $f_n=e^{\log(f_n)}$, we should get $$f_n= {n^{1/8}}\left(1+\frac{3}{32 n}-\frac{51}{2048 n^2}+\frac{405}{65536 n^3}+O\left(\frac{1}{n^4}\right) \right)$$ which, for sure, shows the limit.

Concerning the values for small $n$, the approximation gives $10$ exact significant figures as soon as $n > 22$.