  As is well known, the idea of a fractional sum and difference on uniform
lattice is more current, and gets a lot of development in this field. But the
definitions of fractional sum and fractional difference of $f(z)$ on
non-uniform lattices $x(z)=c_{1}z^{2}+c_{2}z+c_{3}$ or
$x(z)=c_{1}q^{z}+c_{2}q^{-z}+c_{3}$ seem much more difficult and complicated.
In this article, for the first time we propose the definitions of the
fractional sum and fractional difference on non-uniform lattices by two
different ways. The analogue of Euler's Beta formula, Cauchy' Beta formula on
on non-uniform lattices are established, and some fundamental theorems of
fractional calculas, the solution of the generalized Abel equation and
fractional central difference equations on non-uniform lattices are obtained
etc.
