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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·(*)

The authors consider

f n = 2 F 1 [a+ε 1 n,b+ε 2 n;c+ε 3 n;z],ε 1 ,ε 2 ,ε 3 {-1,0,1}·

From elementary properties of 2 F 1 it is shown that out of the 26 non-zero triples (ε 1 ,ε 2 ,ε 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 (*) 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 [2 3,1;4 3+n;exp(1 3πi)]· 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 (*)· The result agrees well with the exact value of f 0 in terms of Γ(1 3) and Γ(2 3)·

MSC:
33C05Classical hypergeometric functions, 2 F 1
39A11Stability of difference equations (MSC2000)
65D20Computation of special functions, construction of tables