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Zbl 0648.39002
Gray, Henry L.; Zhang, Nien fan
On a new definition of the fractional difference.
(English)
[J] Math. Comput. 50, No.182, 513-519 (1988). ISSN 0025-5718; ISSN 1088-6842/e

Differences of fractional order are introduced so that they obey a general exponential law $\Delta\sp{r+s}=\Delta\sp r\Delta\sp s$, 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 $\nabla\sp{t}\sb{a} {}\sp{\alpha}$ of order $\alpha$ here defined has the nice'' properties, like linearity, composition of the form $\nabla\sp{t}\sb{a} {}\sp{\alpha} \nabla\sp{t}\sb{a} {}\sp{\beta}= \nabla\sp{t}\sb{a} {}\sp{\alpha +\beta}$, $\beta\ne 1,2,..$. (for positive integers $\beta$ a more complicated formula is given) the Leibniz rule $$\nabla\sp{t}\sb{a} {}\sp{\alpha}f(t)g(t)= \sum\sp{t- a}\sb{n=0}\binom{\alpha}{n} [\nabla\sp{t-n}\sb{a} {}\sp{\alpha-n}f(t-n)] [\nabla\sp n g(t)].$$ 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 $\alpha$.
[J.Gregor]
MSC 2000:
*39A70 Difference operators
39A12 Discrete version of topics in analysis
47B39 Difference operators (operator theory)

Keywords: fractional difference; backward difference operator; Leibniz rule; Differences of fractional order; difference operator; second order linear difference equation with variable coefficients

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