The asymptotic behavior of hypergeometric function around -1

Question

Recently, in studing some specific orthogonal polynomials on unit circle, I was lead to study the asymptotic behavior of the following hypergeometric function at the neighberhood of $-1$:

$$ f_n(e^{i \theta}) = _2\!\!F_1(s, -n; -n-s; e^{i\theta}) = \sum_{k = 0}^n \frac{(s)_k(-n)_k}{(-n-s)_k k!} e^{i k \theta}.$$ What is the asymptotic behavior of $ |f_n(e^{i \theta})|$ when $\theta$ is near $\pi$ ? Do we have some estimate as the Szego type inequalities for Jacobi polynomials?

Does some one know any references related to this? Thank you in advance for any kind of suggestions.

Answer 1

I just come across on a paper which answers my own question above. (turns out to be very easy.) Here it is:

By using the transformation of hypergeometric functions \begin{align*} _2F_1\left(\begin{array}{c} a, b \\ c\end{array}; z \right) = (1-z)^{-a} \, _2F_1\left(\begin{array}{c} a, c - b\\ c \end{array}; \frac{z}{z-1} \right),\end{align*} we have for $\big| \frac{z}{z-1} \big| \le r <1,$ the following uniform expansion: \begin{align*} & _2F_1\left( \begin{array}{c} s, -n \\ -n - s \end{array}; z \right) = (1-z)^{-s} {_2F_1}\left( \begin{array}{c} s, -s \\ -n - s \end{array}; \frac{z}{z-1} \right) \\ = & (1-z)^{-s} \left[ 1 + \frac{s(-s)}{-n - s} \frac{z}{z-1} + \frac{s(s+1) (-s) (-s + 1)}{ (-n - s) (-n -s +1)} \left(\frac{z}{z-1}\right)^2 + \cdots \right] \\ = & (1 - z)^{-s}\left[1 + \mathcal{O} \left(\frac{1}{n}\right)\right]. \end{align*}