## 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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