# An asymptotic behavior of a sequence of special polynomials

For $$n\to\infty$$, I would like to know the asymptotic behavior of the polynomials defined in terms of the Gauss hypergeometric series: $$p_{n}(z):={}_{2}F_{1}(-n,-nz+\alpha;1;\beta),$$ where $$\alpha,\beta>0$$ are parameters and $$z\in\mathbb{C}$$.

Remarks:

1. I would like to stress that $$z$$ is complex and not only real.

2. My guess is that the asymptotic behavior of the function $${}_{2}F_{1}$$ has already been studied in various settings in literature. I just cannot find the right reference. For example, I am aware of DLMF and the series of papers "Uniform asymptotic expansions for hypergeometric functions with large parameters I-IV" written by Daalhuis etal.

3. I've tried to deduce the asymptotic expansion by applying the saddle-point method to an integral representation of $$p_{n}$$. I think I found a correct formula but I was not able to rigorously justify the saddle-point method (to verify the existence of the steepest descent path). Particularly, the asymptotic behavior depend on the location of $$z$$ in the complex plane.

Any comment on an applicable method or a possible reference would be very appreciated!

• Please give us the formula you have found. – Somos Apr 3 at 16:23