Asymptotics of the hypergeometric function (Q2725064)

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 NEWLINE\[NEWLINE\begin{aligned} &_2F_1\left[ a+ \lambda, b-\lambda; c;{1-z\over 2}\right]\\ &\sim\Gamma (c)2^{(a+b-1)/2}(z-1)^{-c/2}(z+1)^{(c-a-b-1)/2} \left({\sinh \zeta \over \zeta}\right)^{1/2}\times\\ &\times \alpha^{1-c} \left\{\zeta I_{c-1}(\alpha \zeta) \sum_s{A_s (\zeta)\over \alpha^{2s}} +\zeta^2 I_{c-2}(\alpha \zeta) \sum_s{B_s (\zeta)\over \alpha^{2s +1}}\right\},\\ &|\lambda |\to\infty,\;|\arg \lambda|<\pi, \end{aligned}NEWLINE\]NEWLINE 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.