I'm reading about Legendre polynomials for additional information since it is interesting to know! Moreover it would help me with a task I am working on. See https://math.stackexchange.com/questions/3945490

The generating function of Legendre polynomials $P_n(x)$ is defined as $$\frac{1}{\sqrt{1-2tx+t^2}}=\sum\limits_{n=0}^{\infty}P_n(x)t^n.$$

It is known that for large $n$ the asymptotic expansion of the Legendre polynomials is

$$P_n(x) \sim \frac{1}{\sqrt{2\pi n}}\frac{(x+(x^2-1)^{1/2})^{n+1/2}}{(x^2-1)^{1/4}}.$$

My question is how to prove this, without using integrals (if there exists such a proof). I searched a lot, but all the proofs I found regarding this start from the integral representation of the Legendre polynomials. Is it possible to prove it by starting directly from the generating function?

Thanks!