Does anyone know where to find (or how to obtain) expressions for the Legendre functions for large degree, to second order? For example, to first order the expressions are $$ P_n(\cosh(x)) ~ \substack{\huge\rightarrow\\\scriptstyle{n\rightarrow\infty}}~ \frac{1}{\sqrt{\pi n}}\frac{e^{(n+1/2)x}}{\sqrt{2\sinh(x)}} \\ Q_n(\cosh(x))~ \substack{\huge\rightarrow\\\scriptstyle{n\rightarrow\infty}}~ \sqrt{\frac{\pi}{n}}\frac{e^{-(n+1/2)x}}{\sqrt{2\sinh(x)}}, $$ for integer $n$. These are apparently given on page 305-306 of "The theory of spherical and ellipsoidal harmonics" by E. W. Hobson, although I can't seem to access this book.

I am looking for the second order of these expressions - corrections $\mathcal{O}(1/n)$ relative to the above expressions.