Suppose I know that the generating function of some sequence $(f_n)_n$ is $$ G(f_n;x) = \exp\left[-\alpha\cdot\arcsin(\beta\cdot x)+\frac{1}{2}\log\left(\gamma\sqrt{1-\delta \cdot x^2}+1-\delta \cdot x\right)\right] $$ for some constants $\alpha,\beta,\gamma,\delta$. How can one deduce the asymptotic formula of the coefficients corresponding to the above generating function?
It seems that usual techniques involving Tauberian theorem, or the "Asymptotic growth of a sequence" section in
https://en.wikipedia.org/wiki/Generating_function
are not so helpful. Is there any way to do that?