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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 $(1)\quad D\sp nx\sp{\lambda}=\lambda (\lambda -1)...(\lambda -n+1)x\sp{\lambda -n},$ 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 $(2)\quad \Delta\sp nf(x)=\sum\sp{n}\sb{k=0}(-1)\sp k\left( \matrix n\\ k\endmatrix \right)f(x+n-k),$ 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

39A70Difference operators
39A12Discrete version of topics in analysis
26A33Fractional derivatives and integrals (real functions)