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Zbl 1117.33005
Gil, Amparo; Segura, Javier; Temme, Nico M.
Numerically satisfactory solutions of hypergeometric recursions. (English)
[J] Math. Comput. 76, No. 259, 1449-1468 (2007). ISSN 0025-5718; ISSN 1088-6842/e

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.\par 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.
[Francisco Perez Acosta (La Laguna)]

MSC 2000:: *33C05 Classical hypergeometric functions
39A11 Stability of difference equations
41A60 Asymptotic problems in approximation
65D20 Computation of special functions

Keywords: Gauss hypergeometric functions; recursion relations; difference equations; stability of recursion relations; numerical evaluation of special functions; asymptotic analysis

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