Each family of Gauss hypergeometric functions $$ f_n={}_2F_1(a+\varepsilon_1n, b+\varepsilon_2n ;c+\varepsilon_3n; z),\quad n\in {\mathbb Z}\,, $$ for fixed $\varepsilon_j=0,\pm1$ (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.

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Mathematics of Computation

Gil, A, Segura, J, & Temme, N.M. (2007). Numerically satisfactory solutions of hypergeometric recursions. Mathematics of Computation, 76(259), 1449–1468.