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The ABC of hyper recursions. (English) Zbl 1092.33004
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 $\Gamma (\frac{2}{3}).$

33C05Classical hypergeometric functions, ${}_2F_1$
39A11Stability of difference equations (MSC2000)
65D20Computation of special functions, construction of tables
Full Text: DOI arXiv
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