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 \(\left[ { - 1,1} \right]\).

中文翻译：

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关于指数在x = 1附近的勒让德多项式的渐近展开：Mehler-Rayleigh公式的推广

摘要

勒让德多项式\（{{P} _ {n}} \ left（x \ right）\）的在\（x = 1 \）附近的指数\（n \）的反幂的渐近展开是获得。证明\（{{n} ^ {{--k}}} \）的扩展系数是形式为\（{{rho} ^ {p}} {{J} _ {p}} \ left（\ rho \ right）\），其中\（0 \ leqslant p \ leqslant k \）。还显示了扩展的第一项与区间\（-1 \ leqslant x \ leqslant 1 \）端点附近附近的勒让德多项式的公知扩展一致在中间极限。基于此结果，可以在整个区间\（\ left [{-1,1} \ right] \）中获得Legendre多项式相对于索引的均匀展开。