Evaluation of a generalized hypergeometric function

Question

Is there some known identity that can simplify the following evaluation of the generalized hypergeometric function: $$_{2}F_{3}\left(1,n+\frac{1}{2};n+1,n+1,n+\frac{3}{2};1\right)$$ where $n\geq 2$ is a positive integer. By simplification, I mean a functional expression in terms of $n$, not a series sum. If there is known asymptotic for large $n$, I will be interested in that too.

Answer 1

For a closed-form in terms of ratios of gamma functions, you need, for ${}_pF_q,$ p=q+1. An asymptotic expansion is not difficult to derive, however. $$G(n)={}_2F_3(1,n-1/2;n,n,n+1/2;1)=1+\sum_{k=1}^\infty \tfrac{n-1/2}{n-1/2+k} \frac{1}{((n)_k)^2} .$$ The summand is not oscillating, so expand the summand as $n \to \infty,$ $$G(n)\sim 1+ \sum_{k=1}^\infty x^k \big(1-k^2/n + O(n^{-2}) \big) \ , \ x=n^{-2}$$ $$ \sim \frac{1}{1-x}\big(1-\frac{1}{n} \frac{x(1+x)}{(1-x)^2} \big).$$

As an example, the difference between $G(n)$ (offset by 1 from the proposer's problem) and the asymptotic expansion is only 0.0056% for $n$ as small as 10.