The ABC of Hyper Recursions

Abstract

Each family of Gauss hypergeometric functions $$ f_n={}_2F_1(a+ε_1n, b+ε_2n ;c+ε_3n; z), $$ for fixed $ε_j=0,\pm1$ (not all $ε_j$ equal to zero) satisfies a second order linear difference equation of the form $$ A_nf_{n-1}+B_nf_n+C_nf_{n+1}=0. $$ Because of symmetry relations and functional relations for the Gauss functions, many of the 26 cases (for different $ε_j$ 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 $f_n$ (minimal or dominant) as $n\to \pm\infty$. A second independent solution is given in each basic case which is dominant when $f_n$ 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.