Abstract
Properties of discrete fractional calculus in the sense of a backward difference are introduced and developed. Exponential laws and a product rule are developed and relations to the forward fractional calculus are explored. Properties of the Laplace transform for the nabla derivative on the time scale of integers are developed and a fractional finite difference equation is solved with a transform method. As a corollary, two new identities for the gamma function are exhibited.
Highlights
In this article, we shall continue our recent work to study discrete fractional calculus
There has been little work done in the study of discrete fractional calculus in the case of the forward difference
There has been more work in the study of discrete fractional calculus using the backward difference [11, 8, 10, 12, 13]; applications arising in time series analysis have motivated the development in discrete backward fractional calculus
Summary
We shall continue our recent work to study discrete fractional calculus. We shall continue to employ terminology from the theory of time scales calculus [5] and in this article, we shall employ the backward difference, or the nabla derivative. There has been little work done in the study of discrete fractional calculus in the case of the forward difference. Miller and Ross [15] initiated the study and the authors [2, 3, 4] have more recently been developing discrete forward fractional calculus. Throughout this article, we shall compare and contrast the methods and results with the analogous methods and results for fractional forward difference calculus
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