An analytic continuation of the hypergeometric series
SIAM Journal on Mathematical Analysis
Computing the hypergeometric function
Journal of Computational Physics
Computing toroidal functions for wide ranges of the parameters
Journal of Computational Physics
Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
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Each member of the family of Gauss hypergeometric functions fn=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 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 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.