The ABC of hyper recursions

Each member of the family of Gauss hypergeometric functions f n = 2 F 1 ( a + ε 1 n , b + ε 2 n ; c + ε 3 n ; z ) , where a , b , c and z do not depend on n, and ε j = 0 , ± 1 (not all ε j equal to zero) satisfies a second order linear difference equation of the form A n f n - 1 + B n f n + C n f n + 1 = 0 . Because of symmetry relations and functional relations for the Gauss functions, the set of 26 cases (for different ε j values) can be reduced to a set of 5 basic forms of difference equations. In this paper the coefficients A n , B n and C n of these basic forms are given. In addition, domains in the complex z-plane are given where a pair of minimal and dominant solutions of the difference equation have to be identified. The determination of such a pair asks for a detailed study of the asymptotic properties of the Gauss functions f n for large values of n, and of other Gauss functions outside this group. This will be done in a later paper.