On the Definitions of Fractional Sum and Difference on Non-uniform Lattices

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.