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Sum and difference compositions in discrete fractional calculus. (English) Zbl 1248.39003

The author introduces in the \(\nu^{\text{th}}\)-order fractional sum of \(f:\mathbb{N}_{a}\to\mathbb{R}\) as \[ \left(\Delta^{-\nu}_{a}f\right)\!\!(t):=\begin{cases}\frac{1}{\Gamma(\nu)}\sum_{s=a}^{t-\nu}\left(t-\sigma(s)\right)\!\!\frac{\nu-1}{\,}\! f(s),&\quad \nu>0,\;t\in\mathbb{N}_{a+\nu}:=\left\{a+\nu,a+1+\nu,\dots\right\}\\ f(t),&\quad \nu=0,\;t\in\mathbb{N}_a,\end{cases} \] and the \(\nu^{\text{th}}\)-order fractional difference as \[ (\Delta^{\nu}_{a} f)(t):=\Delta^{N}\Delta_{a}^{-(N-\nu)}f(t), \quad t\in\mathbb{N}_{a+N-\nu}, \] where \(\nu\geq 0\) is given and \(N\in\mathbb{N}\) is such that \(N-1<\nu\leq N\). In Section 2, the basic properties of the fractional sum and difference operators are established. Section 3 is devoted to the study of composition rules for these operators (all possible combinations are discussed). In Section 4, the existence of the solution for the \(\nu^{\text{th}}\)-order fractional difference equation \[ \Delta^{\nu}_{a+\nu-N} y(t)=f(t), \quad t\in\mathbb{N}_a, \] and the uniqueness of the solution for the latter equation completed with the initial values \[ \Delta^i y(a+\nu-N)=A_i,\quad i\in\{0,1,\dots,N-1\},\;A_i\in\mathbb{R}, \] are established. Finally, the developed theory is illustrated by two examples.

MSC:

39A12 Discrete version of topics in analysis
26A33 Fractional derivatives and integrals
34A08 Fractional ordinary differential equations
39A10 Additive difference equations
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