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On a new definition of the fractional difference. (English) Zbl 0648.39002

Differences of fractional order are introduced so that they obey a general exponential law ${{\Delta }}^{r+s}={{\Delta }}^{r}{{\Delta }}^{s}$, the Leibniz rule and in addition, the definition involves only finite summation. This is achieved through a generalization of the binomial coefficient and by “inverting” the operator of the n-fold summation. The order $\alpha$ can be any complex number and only values of f at integer points are involved. It is proved that the difference operator ${\nabla }_{a}^{t}{}^{\alpha }$ of order $\alpha$ here defined has the “nice” properties, like linearity, composition of the form ${\nabla }_{a}^{t}{}^{\alpha }{\nabla }_{a}^{t}{}^{\beta }={\nabla }_{a}^{t}{}^{\alpha +\beta }$, $\beta \ne 1,2,··$. (for positive integers $\beta$ a more complicated formula is given) the Leibniz rule

${\nabla }_{a}^{t}{}^{\alpha }f\left(t\right)g\left(t\right)=\sum _{n=0}^{t-a}\left(\genfrac{}{}{0pt}{}{\alpha }{n}\right)\left[{\nabla }_{a}^{t-n}{}^{\alpha -n}f\left(t-n\right)\right]\left[{\nabla }^{n}g\left(t\right)\right]·$

Further results and applications are also given. It is shown that the solution of a second order linear difference equation with variable coefficients can be expressed in terms of the coefficients using differences of fractional order. The limiting case of the differences introduced here are proved to be identical to earlier considered differences of arbitrary real order $\alpha$.

Reviewer: J.Gregor
##### MSC:
 39A70 Difference operators 39A12 Discrete version of topics in analysis 47B39 Difference operators (operator theory)