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Zbl 0979.33002
Jones, D.S.
Asymptotics of the hypergeometric function. (English)
[J] Math. Methods Appl. Sci. 24, No.6, 369-389 (2001). ISSN 0170-4214; ISSN 1099-1476/e

The paper deals with $_2F_1[a+ \lambda,b-\lambda; c;{1\over 2} (1-z)]$ for large values of $|\lambda|$. Following Olver, the author transforms the differential equation such that solutions suitable for asymptotic investigation emerge. Auxiliary variables $z=\text {cosh} \zeta$, $\alpha= {1 \over 2}(a-b)+ \lambda$, are introduced, and $\alpha$ 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 $-\infty$ to $-1$, and $c$ is real, then $$\align &_2F_1\left[ a+ \lambda, b-\lambda; c;{1-z\over 2}\right]\\ &\sim\Gamma (c)2^{(a+b-1)/2}(z-1)^{-c/2}(z+1)^{(c-a-b-1)/2} \left({\sinh \zeta \over \zeta}\right)^{1/2}\times\\ &\times \alpha^{1-c} \left\{\zeta I_{c-1}(\alpha \zeta) \sum_s{A_s (\zeta)\over \alpha^{2s}} +\zeta^2 I_{c-2}(\alpha \zeta) \sum_s{B_s (\zeta)\over \alpha^{2s +1}}\right\},\\ &|\lambda |\to\infty,\ |\arg \lambda|<\pi, \endalign$$ where $I_\nu$ denotes a modified Bessel function, and the coefficient functions $(A_s(\zeta))$ and $(B_s(\zeta))$ satisfy certain differential-recursion equations. The author's expansion has a wider range of validity than an expansion in powers of $1/\lambda$ given by Watson. Further results based upon transformations of $_2F_1$ are considered. Also, results involving Legendre functions are noted as particular cases. Finally, the author derives an expansion where $\{\cdots+\cdots\}$ is replaced with $\sum_mC_m (\zeta) \zeta I_{c-1+m} (\alpha\zeta) \alpha^{-m}$. The coefficient functions $(C_m (\zeta))$ again satisfy a differential-recursion equation.
[Per W.Karlsson (Lyngby)]

MSC 2000:: *33C05 Classical hypergeometric functions
41A60 Asymptotic problems in approximation

Keywords: Olver's method

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