Fractional sum and fractional difference on non-uniform lattices and analogue of Euler and Cauchy Beta formulas

Abstract

As is well known, the definitions of fractional sum and fractional difference of f (z) on non-uniform lattices x(z) = c₁z² + c₂z + c₃ or x(z) = c₁q^z + c₂q^−z + c₃ are 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 non-uniform lattices are established, and some fundamental theorems of fractional calculas, the solution of the generalized Abel equation on non-uniform lattices are obtained etc.