Asymptotic Expansion of Legendre Polynomials with Respect to the Index near x = 1: Generalization of the Mehler–Rayleigh Formula

Abstract

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 $\left[ { - 1,1} \right]$.