Suppose $a,b,c,d,e\in\mathbb{R}$ are such that $d+e-a-b-c>0$ and $d,e\notin\mathbb{Z}$. I would like to know whether it is possible to deduce an asymptotic formula for the sequence given by the hypergeometric function $$ {}_{3}F_{2}(a,b-n,c-n;d-n,e-n;1), \quad \mbox{ for }\; n\to\infty. $$ Let me remark that I have checked DLMF as well as monographs on special functions by Bailey; Prudnikov, Brychkov, Marichev; and Erdelyi but have not found the answer.
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$\begingroup$ Did you try to use the three-term transformation ((3.1.3) in Gasper and Rahman, Basic hypergeometric series)? Then you will get a sum of two ${}_3F_2$-series that only have two parameters depending on $n$ instead of four. Formally they tend to summable ${}_2F_1$-series when $n\rightarrow\infty$ and perhaps this will allow you to get a hand on the asymptotics. $\endgroup$– Hjalmar RosengrenCommented Sep 17, 2021 at 8:59
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$\begingroup$ @HjalmarRosengren I haven't tried this approach. Thank you for the comment. $\endgroup$– TwiCommented Sep 20, 2021 at 10:21
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