# Asymptotic expansion of hypergeometric function ${}_3F_2$ for large parameters

I encountered the following hypergeometric function in my research: $${}_3F_2(2,1+n,1+n;1,2+n;z)$$ where $$0. I'm interested in its behavior for large $$n$$. Semilog plot suggests exponential increase in $$n$$, however, I'm having trouble deriving the expression for the asymptotic expansion. There are lots of references to 1969 volume by Yudell Luke. I scanned it to no avail for a result that fits the above formula. I am a mere computer scientist, and am unfamiliar with literature on hypergeometric functions (which seems quite extensive). I would appreciate any help.

Below is a closed-form evaluation in terms of simple functions. Let $$y=z/(z-1).$$ Then $${}_3F_2(2,n,n;1,n+1;z)=n(-z)^{-n}\Big((n-1)\big(\log(1-y)+\sum_{k=1}^{n-1}\frac{y^k}{k} \big) + y^n \Big)$$ I proved this by using Pochhammer symbol properties, which gets me to a linear combination of $${}_2F_1.$$ Then I used a linear transformation to get from the argument $$z$$ to $$y.$$ That series can be manipulated to give the logarithm and a finite sum. Variable $$y$$ is always negative, but if it is small, the sum will converge rapidly. If you put this on a computer, watch out for large negative $$y,$$ which occurs for $$z$$ close to 1. In the finite sum, I'd probably add terms pairwise. If the $$z\sim 1$$ case is your most important case, then it's probably worth thinking about this some more.
Added: The sum for $$z\sim 1$$ can be done by an expansion found in 'Asymptotic Expansions Pertaining to the Logarithmic Series and Related Trigonometric Sums,' G. Fikioris & P. Andrianesis, J. Class. Analysis vol 7 #2, (2015) 113-127. $$\sum_{k=1}^{n-1}\frac{y^k}{k} \sim y^n\sum_{k=0}^\infty \frac{A_k(y)}{(y-1)^{k+1}} \frac{1}{n^{k+1}} \, ,n \to \infty$$ where the $$A_k(y)$$ are the Eulerian polynomials and begin with $$A_0(y)=1$$ and $$A_1(y) = y.$$