  A general explicit form for generating functions for approximating fractional
derivatives is derived. To achieve this, an equivalent characterisation for
consistency and order of approximations established on a general generating
function is used to form a linear system of equations with Vandermonde matrix
for the coefficients of the generating function which is in the form of power
of a polynomial. This linear system is solved for the coefficients of the
polynomial in the generating function. These generating functions completely
characterise Gr\"unwald type approximations with shifts and order of accuracy.
Incidentally, the constructed generating functions happen to be generalization
of the previously known Lubich forms of generating functions without shift. As
a consquence, a general explicit form for new finite difference formulas for
integer-order derivatives with any order of accuracy are derived.
