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Numerically satisfactory solutions of hypergeometric recursions. (English) Zbl 1117.33005
Each family of Gauss hypergeometric functions $$f_n= {_2F_1}(a+\varepsilon_1 n,b+\varepsilon_2 n, c+\varepsilon_3 n),\quad n\in\bbfN,$$ for fixed $\varepsilon_j= 0,\pm1$ $(\varepsilon^2_1+ \varepsilon^2_2+ \varepsilon^2_3\ne 0)$ satisfies a second-order linear difference equation of the form $$\Delta_n f_{n-1}+ B_n f_n+ C_n f_{n+1}= 0.$$ 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 Poincaré 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.

33C05Classical hypergeometric functions, ${}_2F_1$
39A11Stability of difference equations (MSC2000)
41A60Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
65D20Computation of special functions, construction of tables
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