*(English)*Zbl 1117.33005

Each family of Gauss hypergeometric functions

for fixed ${\epsilon}_{j}=0,\pm 1$ $({\epsilon}_{1}^{2}+{\epsilon}_{2}^{2}+{\epsilon}_{3}^{2}\ne 0)$ satisfies a second-order linear difference equation of the form

Only with four basic difference equations can all the other 26 cases be obtained by symmetry relations and functional relations. For each of these recurrences, the authors give pairs of numerically satisfactory solutions in the regions in the complex plane $|{t}_{1}|\ne |{t}_{2}|$, ${t}_{1}$, ${t}_{2}$ being the roots of the characteristic equation. This is an essential piece of information for the computation of hypergeometric functions by means of recurrence relations.

In the case of the critical curves $|{t}_{1}|=|{t}_{2}|$ the PoincarĂ© theorem does not provide information regarding the existence of minimal solutions. The study of the behaviour on the critical curves needs an separate analysis and is beyong the scope of the present paper.

##### MSC:

33C05 | Classical hypergeometric functions, ${}_{2}{F}_{1}$ |

39A11 | Stability of difference equations (MSC2000) |

41A60 | Asymptotic approximations, asymptotic expansions (steepest descent, etc.) |

65D20 | Computation of special functions, construction of tables |