The ABC of hyper recursions

摘要

Each member of the family of Gauss hypergeometric functionsfn=2F1(a+ε1n,b+ε2n;c+ε3n;z),where a,b,c and z do not depend on n, and εj=0,±1 (not all εj equal to zero) satisfies a second order linear difference equation of the formAnfn-1+Bnfn+Cnfn+1=0.Because of symmetry relations and functional relations for the Gauss functions, the set of 26 cases (for different εj 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.