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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!

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

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