*(English)*Zbl 1092.33004

Some of the three-term relations between associated Gaussian hypergeometric functions are also second-order linear difference equations of the form

The authors consider

From elementary properties of ${}_{2}{F}_{1}$ it is shown that out of the 26 non-zero triples $({\epsilon}_{1},{\epsilon}_{2},{\epsilon}_{3})$ only five have to be considered. In these cases, a number of details are given, notably ${A}_{n},{B}_{n},{C}_{n}$ in terms of $a,b,c,z;$ also, second solutions to $(*)$ are obtained by the aid of the transformations to functions of $1-z$ and $1/z\xb7$ Moreover, numerical aspects (e.g., stability) are discussed. As an example, they consider ${f}_{n}={\phantom{\rule{0.166667em}{0ex}}}_{2}{F}_{1}[\frac{2}{3},1;\frac{4}{3}+n;exp\left(\frac{1}{3}\pi \text{i)}\right]\xb7$ The power series is ill-suited for computation of ${f}_{0}$ but it works well for ${f}_{29}$ and ${f}_{30};$ and from these values ${f}_{0}$ is eventually obtained by backward recursion by means of $(*)\xb7$ The result agrees well with the exact value of ${f}_{0}$ in terms of ${\Gamma}\left(\frac{1}{3}\right)$ and ${\Gamma}\left(\frac{2}{3}\right)\xb7$

##### MSC:

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

39A11 | Stability of difference equations (MSC2000) |

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