## Fractional difference calculus.(English)Zbl 0693.39002

Univalent functions, fractional calculus, and their applications, 139-152 (1989).
[For the entire collection see Zbl 0683.00012.]
In 1886 Laurent treated the well-known formula $$(1)\quad D^ nx^{\lambda}=\lambda (\lambda -1)...(\lambda -n+1)x^{\lambda -n},$$ where n is a positive integer, and defined the meaning of this formula (1) when n is not necessarily a positive integer. In the present paper the authors treat the well-known formula of difference operators $$(2)\quad \Delta^ nf(x)=\sum^{n}_{k=0}(-1)^ k\left( \begin{matrix} n\\ k\end{matrix} \right)f(x+n-k),$$ where n is a positive integer, and define the meaning of (2) when n is not necessarily a positive integer. Furthermore, they derive some consequences of this definition and present its application.
Reviewer: H.Haruki

### MSC:

 39A70 Difference operators 39A12 Discrete version of topics in analysis 26A33 Fractional derivatives and integrals

### Keywords:

fractional difference calculus; difference operators

Zbl 0683.00012