Higher order geometric difference operators

Standard central difference operators are typically very poor at approximating derivatives in the vicinity of singularities. In such cases geometric sequences have been used successfully as the basis for defining difference operators up to second order which give excellent approximation. In this paper, geometric difference methods are used to find operators of all orders n which give exact higher derivatives for functions of the form αx^-1 + Σ_m=0ⁿ ß_mx^m. Surprisingly good analogies are found with classical results and some numerical and computational comparisons are made.