I came across the following hypergeometric function recently: $$ _2F_1(1-n,p-2n+1;p-n+1;x) $$ where $p > 0$ is a non-integer constant, $n$ some large positive integer, and $x > 0$ a small constant.

I'd like to know its growth order. I am expecting it to decay like $(1-x)^{2n}$ or so but I have no idea how to prove it. Invoking the identity $$_2F_1(a,b;c;x) = (1-x)^{c-a-b} {}_2F_1(c-a,c-b;c;x)$$ I get $$ _2F_1(1-n,p-2n+1;p-n+1;x) = (1-x)^{2n-1} {}_2F_1(p,n;p-n+1;x) $$ so I am expecting that when $n$ is sufficiently large, $_2F_1(p,n;p-n+1;x)$ grows very slowly (with respect to $n$), slower than $n^{p-\epsilon}$ for some small $\epsilon$ that may depend on $x$ and $p$. I tried numerical values in Mathematica, and it seems that $_2F_1(p,n;p-n+1;x)$ is positive, slowly increasing and upper bounded by $1$ when $n$ is sufficiently large. In fact, proving it is upper bounded by some constant (could depend on $p$ and $x$) would be already good for me.

I tried to expand it into series and compared the term of the same index for $n$ and $n+1$, but this did not work.

If it is well-known, can anyone point me to some reference?

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