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Asymptotics of the hypergeometric function. (English) Zbl 0979.33002

The paper deals with 2 F 1 [a+λ,b-λ;c;1 2(1-z)] for large values of |λ|. Following Olver, the author transforms the differential equation such that solutions suitable for asymptotic investigation emerge. Auxiliary variables z=coshζ, α=1 2(a-b)+λ, are introduced, and α is taken as the large expansion parameter. As a result of rather long calculations, including a discussion of error bounds, the following result is obtained. If the z-plane is cut from - to -1, and c is real, then

2 F 1 a+λ,b-λ;c;1-z 2Γ(c)2 (a+b-1)/2 (z-1) -c/2 (z+1) (c-a-b-1)/2 sinhζ ζ 1/2 ××α 1-c ζI c-1 (αζ) s A s (ζ) α 2s +ζ 2 I c-2 (αζ) s B s (ζ) α 2s+1 ,|λ|,|argλ|<π,

where I ν denotes a modified Bessel function, and the coefficient functions (A s (ζ)) and (B s (ζ)) satisfy certain differential-recursion equations. The author’s expansion has a wider range of validity than an expansion in powers of 1/λ given by Watson. Further results based upon transformations of 2 F 1 are considered. Also, results involving Legendre functions are noted as particular cases. Finally, the author derives an expansion where {+} is replaced with m C m (ζ)ζI c-1+m (αζ)α -m . The coefficient functions (C m (ζ)) again satisfy a differential-recursion equation.

MSC:
33C05Classical hypergeometric functions, 2 F 1
41A60Asymptotic approximations, asymptotic expansions (steepest descent, etc.)