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Zbl 1092.33004
Gil, Amparo; Segura, Javier; Temme, Nico M.
The ABC of hyper recursions.
(English)
[J] J. Comput. Appl. Math. 190, No. 1-2, 270-286 (2006). ISSN 0377-0427

Some of the three-term relations between associated Gaussian hypergeometric functions are also second-order linear difference equations of the form $$A_{n} f_{n-1}+ B_{n} f_{n}+ C_{n} f_{n+1}=0. \tag*$$ The authors consider $$f_{n}=\,_{2}F_{1}[ a+\varepsilon _{1}n,b+ \varepsilon _{2}n;c+ \varepsilon _{3}n;z] ,\qquad \varepsilon _{1}, \varepsilon _{2},\varepsilon _{3}\in \{ -1,0,1\} .$$ From elementary properties of $_{2}F_{1}$ it is shown that out of the 26 non-zero triples $( \varepsilon _{1},\varepsilon _{2},\varepsilon _{3})$ 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 $( \ast )$ are obtained by the aid of the transformations to functions of $1-z$ and $1/z.$ Moreover, numerical aspects (e.g., stability) are discussed. As an example, they consider $f_{n}=\,_{2}F_{1}[ \frac{2}{3},1;\frac{4}{3}+n;\exp ( \frac{1}{3}\pi \text{i)}] .$ 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 $( \ast ) .$ The result agrees well with the exact value of $f_{0}$ in terms of $\Gamma (\frac{1}{3})$ and $\Gamma (\frac{2}{3}).$
[Per W. Karlsson (Lyngby)]
MSC 2000:
*33C05 Classical hypergeometric functions
39A11 Stability of difference equations
65D20 Computation of special functions

Keywords: Hypergeometric functions; Difference equations; Recursion relations; Numerical evaluation of special functions

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