Numerically satisfactory solutions of hypergeometric recursions

Abstract:

Each family of Gauss hypergeometric functions \[ f_n={}_2F_1(a+\varepsilon _1n, b+\varepsilon _2n ;c+\varepsilon _3n; z), n\in {\mathbb Z} , \] for fixed $\varepsilon _j=0,\pm 1$ (not all $\varepsilon _j$ equal to zero) satisfies a second order linear difference equation of the form \[ A_nf_{n-1}+B_nf_n+C_nf_{n+1}=0. \] Because of symmetry relations and functional relations for the Gauss functions, many of the 26 cases (for different $\varepsilon _j$ values) can be transformed into each other. In this way, only with four basic difference equations can all other cases be obtained. For each of these recurrences, we give pairs of numerically satisfactory solutions in the regions in the complex plane where $|t_1|\neq |t_2|$, $t_1$ and $t_2$ being the roots of the characteristic equation.