The ABC of hyper recursions

Each member of the family of Gauss hypergeometric functions fn=2F1(a + e1n, b + e2n; c + e3n; z), where a, b, c and z do not depend on n, and ej = 0, ±1 (not all ej equal to zero) satisfies a second order linear difference equation of the form Anfn-1 + Bnfn + Cnfn+1 = 0. Because of symmetry relations and functional relations for the Gauss functions, the set of 26 cases (for different ej values) can be reduced to a set of 5 basic forms of difference equations. In this paper the coefficients An, Bn and Cn 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 fn for large values of n, and of other Gauss functions outside this group. This will be done in a later paper.