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Error bounds for a uniform asymptotic expansion of the Legendre function ${P}_{n}^{-m}\left(cosh\phantom{\rule{0.166667em}{0ex}}z\right)$. (English) Zbl 0655.33004
The Legendre function ${P}_{n}^{-m}\left(\text{coh}\phantom{\rule{0.277778em}{0ex}}z\right)$ is considered for large values of $n$. The asymptotic expansion is in terms of the modified Bessel function ${I}_{m}\left(z\right)$, and holds uniformly with respect to $z\in \left[0,\infty \right)$; the parameter $m$ is fixed, $m>-1/2$. The method is based on an integral representation of the Legendre function. A recurrence relation is derived for the coefficients in the expansion, and computable error bounds are derived for the remainders. A comparison is given of the new expansion with an earlier expansion given by Olver, especially with respect to the bounds for the remainders.
Reviewer: N.M.Temme
##### MSC:
 33C55 Spherical harmonics 41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
##### Keywords:
Legendre function