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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