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Asymptotics of the hypergeometric function. (English) Zbl 0979.33002

The paper deals with ${}_{2}{F}_{1}\left[a+\lambda ,b-\lambda ;c;\frac{1}{2}\left(1-z\right)\right]$ for large values of $|\lambda |$. Following Olver, the author transforms the differential equation such that solutions suitable for asymptotic investigation emerge. Auxiliary variables $z=\text{cosh}\zeta$, $\alpha =\frac{1}{2}\left(a-b\right)+\lambda$, 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

$\begin{array}{cc}& {}_{2}{F}_{1}\left[a+\lambda ,b-\lambda ;c;\frac{1-z}{2}\right]\hfill \\ & \sim {\Gamma }\left(c\right){2}^{\left(a+b-1\right)/2}{\left(z-1\right)}^{-c/2}{\left(z+1\right)}^{\left(c-a-b-1\right)/2}{\left(\frac{sinh\zeta }{\zeta }\right)}^{1/2}×\hfill \\ & ×{\alpha }^{1-c}\left\{\zeta {I}_{c-1}\left(\alpha \zeta \right)\sum _{s}\frac{{A}_{s}\left(\zeta \right)}{{\alpha }^{2s}}+{\zeta }^{2}{I}_{c-2}\left(\alpha \zeta \right)\sum _{s}\frac{{B}_{s}\left(\zeta \right)}{{\alpha }^{2s+1}}\right\},\hfill \\ & |\lambda |\to \infty ,\phantom{\rule{4pt}{0ex}}|arg\lambda |<\pi ,\hfill \end{array}$

where ${I}_{\nu }$ denotes a modified Bessel function, and the coefficient functions $\left({A}_{s}\left(\zeta \right)\right)$ and $\left({B}_{s}\left(\zeta \right)\right)$ 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 ${}_{2}{F}_{1}$ are considered. Also, results involving Legendre functions are noted as particular cases. Finally, the author derives an expansion where $\left\{\cdots +\cdots \right\}$ is replaced with ${\sum }_{m}{C}_{m}\left(\zeta \right)\zeta {I}_{c-1+m}\left(\alpha \zeta \right){\alpha }^{-m}$. The coefficient functions $\left({C}_{m}\left(\zeta \right)\right)$ again satisfy a differential-recursion equation.

##### MSC:
 33C05 Classical hypergeometric functions, ${}_{2}{F}_{1}$ 41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
Olver’s method