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