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Zbl 1055.33008
Dunster, T.M.
Uniform asymptotic expansions for associated Legendre functions of large order. (English)
[J] Proc. R. Soc. Edinb., Sect. A, Math. 133, No. 4, 807-827 (2003). ISSN 0308-2105; ISSN 1473-7124/e

There is an extensive literature on the asymptotic behavior of the associated Legendre functions. This paper fills the gap by treating uniform expansions for large order and small to large degree, both real. The following cases are covered: 1. Fixed $\nu$, complex argument Method: as in [{\it F. W. J. Olver}, Introduction to asymptotics and special functions (1974; Zbl 0308.41023), chapter 12]. 2. Small to large $\nu$, complex argument Method: the Liouville transform from {\it W. G. C. Boyd} and {\it T. M. Dunster} [SIAM J. Math. Anal. 17, 422--450 (1986; Zbl 0591.34048)]. 3. Small to large $\nu$, real argument in $(-1,1)$ for Ferrer's function Method: Liouville transform and results from Olver [loc. cit.]. A well written paper.
[Marcel G. de Bruin (Delft)]

MSC 2000:: *33C45 Orthogonal polynomials and functions of hypergeometric type
41A60 Asymptotic problems in approximation

Keywords: associated Legendre function; Ferrer's function; asymptotic expansion; modified Bessel functions; Liouville-Green type expansion; WKB method

Citations: Zbl 0308.41023; Zbl 0591.34048

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