Numerically satisfactory solutions of hypergeometric recursions

Abstract

Each family of Gauss hypergeometric functions fn = 2F1(a + ε1n,b + ε2n;c + ε3n; z), n ε ℤ, for fixed εj = 0, ±1 (not all εj equal to zero) satisfies a second order linear difference equation of the form Anfn-1 + B nfn + Cnfn+1 = 0. Because of symmetry relations and functional relations for the Gauss functions, many of the 26 cases (for different εj values) can be transformed into each other. In this way, only with four basic difference equations can all other cases be obtained. For each of these recurrences, we give pairs of numerically satisfactory solutions in the regions in the complex plane where |t1| ≠ |t2|, t1 and t2 being the roots of the characteristic equation. ©2007 American Mathematical Society.