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 in inverse powers of the index in a neighborhood of is obtained. It is shown that the expansion coefficient of is a linear combination of terms of the form , where . 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 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 .