Does a limit to this Hypergeometric Function Exist Analytically?

Question

I am interested in evaluating limit $$\lim_{x\rightarrow\pi/2}\left[(\cos x)^n\, _2F_1\left(-\frac{n}{2},-n-m+1;\frac{1}{2}-n;-\frac{16m c}{\cos^2x}\right)\right], $$ where $n$ is a positive even integer, $m$ and $c$ are reals. I checked with Mathematica, and this expression goes to a limiting value at $x\rightarrow \pi/2$, but I want to know if there is any analytical expression interms of $m$ and $c$.

Answer 1

Since $N:=n/2$ is a positive integer, the hypergeometric series terminates and reduces to a polynomial: $$_2F_1\left(-N,b,c,z\right)=\sum_{k=0}^N(-1)^k {N\choose k}\frac{(b)_k}{(c)_k}z^k.$$ As $z\to\infty$, the asymptotics of this expression is determined by the term with $k=N$. Explicitly, $$\lim_{z\to\infty}z^{-N}{}_2F_1\left(-N,b,c,z\right)=(-1)^N\frac{\Gamma(b+N)\Gamma(c)}{\Gamma(c+N)\Gamma(b)}.$$ Setting $b=-n-m+1$, $c=\frac12-n$, $z=-\frac{16mc}{\cos^2x}$ in this formula, we obtain the limit: $$(16mc)^{\frac n2}\frac{\Gamma(-\frac n2-m+1)\Gamma(\frac12-n)}{\Gamma(-n-m+1)\Gamma(\frac12-\frac n2)}.$$