Asymptotic expansion of Legendre polynomials with respect to the index near $x=1$: generalization of the Mehler–Rayleigh formula

An asymptotic expansion of the Legendre polynomials ${{P}_{n}}\left(x\right)$ in inverse powers of the index $n$ in a neighborhood of $x=1$ is obtained. It is shown that the expansion coefficient of ${n}^{{-k}}$ is a linear combination of terms of the form ${{\rho }^{p}}{{J}_{p}}\left(\rho\right)$, where $0\leqslant p\leqslant k$. It is also shown that the first terms of the expansion coincide with a well-known expansion of Legendre polynomials outside neighborhoods of the endpoints of the interval $-1\leqslant x\leqslant 1$ in the intermediate limit. Based on this result, a uniform expansion of Legendre polynomials with respect to the index can be obtained in the entire interval $\left[{-1,1}\right]$.