Zur Asmptotik der hypergeometrischen Funktionen für große Parameterwerte (Asymptotics of hypergeometric functions for large parameters). (German) Zbl 0572.33001
Earlier work of the author on asymptotic expansions for the Gauß hypergeometric function is reconsidered. The type of expansion is
\[
F(a,b;c;z)/\Gamma (c)\sim ((-az)^{-b}/\Gamma (c- b))\sum^{\infty}_{\nu =0}c_{\nu}(b,c,z)(b)_{\nu}a^{-\nu}
\]
\[ +\lambda (1-z)^{c-b-a}((-az)^{b-z}/\Gamma (b))\sum^{\infty}_{\nu =0}c_{\nu}(c-b-c;z/(z-1))(c-b)_{\nu}a^{-\nu}, \] as \(| a| \to \infty\), z fixed, \(b=o(a^{-\delta})\), \(c=o(a^{-\delta})\) \((\delta >0)\). Among others, extension of the earlier results concerns z: the expansion holds for \(| \arg (1-z)| \leq \pi\). Furthermore, recursion relations for the coefficients \(c_{\nu}\) are derived.
\[ +\lambda (1-z)^{c-b-a}((-az)^{b-z}/\Gamma (b))\sum^{\infty}_{\nu =0}c_{\nu}(c-b-c;z/(z-1))(c-b)_{\nu}a^{-\nu}, \] as \(| a| \to \infty\), z fixed, \(b=o(a^{-\delta})\), \(c=o(a^{-\delta})\) \((\delta >0)\). Among others, extension of the earlier results concerns z: the expansion holds for \(| \arg (1-z)| \leq \pi\). Furthermore, recursion relations for the coefficients \(c_{\nu}\) are derived.
Reviewer: M.M.Temme
MSC:
| 33C05 | Classical hypergeometric functions, \({}_2F_1\) |
| 41A60 | Asymptotic approximations, asymptotic expansions (steepest descent, etc.) |
| 30E15 | Asymptotic representations in the complex plane |
