The ABC of Hyper Recursions

Each family of Gauss hypergeometric functions fn = 2F1(a + ε1n, b + ε2n; c + ε3n; z), for fixed 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, many of the 26 cases (for different εj values) can be transformed into each other. We give a set of basic equations from which all other equations can be obtained. For each basic equation, we study the existence of minimal solutions and the character of fn (minimal or dominant) as n → ±∞. A second independent solution is given in each basic case which is dominant when fn is minimal and vice-versa. In this way, satisfactory pairs of linearly independent solutions for each of the 26 second order linear difference equations can be obtained. 2000 Mathematics Subject Classification: 33C05, 39A11, 41A60, 65D20.