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Modeling with fractional difference equations. (English) Zbl 1204.39004

A fractional sum of a function $f$ is introduced as

${{\Delta }}_{a}^{-\alpha }f\left(t\right)=\frac{1}{{\Gamma }\left(\alpha \right)}\sum _{s=a}^{t-\alpha }{\left(t-s-1\right)}^{\left(\alpha -1\right)}f\left(s\right),$

where $a\in R,$ $\alpha >0$, ${x}^{\left(\alpha \right)}={\Gamma }\left(x+1\right)/{\Gamma }\left(x-\alpha +1\right),$ $f$ is defined for $s=a\phantom{\rule{4pt}{0ex}}\left(\text{mod}\phantom{\rule{4.pt}{0ex}}1\right),$ and ${{\Delta }}_{a}^{-\alpha }f$ is defined for $t=a+\alpha \phantom{\rule{4pt}{0ex}}\left(\text{mod}\phantom{\rule{4.pt}{0ex}}1\right)·$ Besides some previously known properties of the fractional sum, additional properties such as a Leibniz type formula and a summation by parts formula are derived. A simple fractional calculus of a variation problem is defined and its Euler-Lagrange equation is derived. As an application, a so called Gompertz fractional difference equation is introduced and solved in terms of a series.

##### MSC:
 39A12 Discrete version of topics in analysis 39A05 General theory of difference equations 26A33 Fractional derivatives and integrals (real functions) 34A08 Fractional differential equations
##### References:
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