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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 $$\left(\Delta^{-\nu}_{a}f\right)\!\!(t):=\cases\frac{1}{\Gamma(\nu)}\sum_{s=a}^{t-\nu}\left(t-\sigma(s)\right)\!\!\frac{\nu-1}{\,}\! f(s),&\quad \nu>0,\ t\in\mathbb{N}_{a+\nu}:=\left\{a+\nu,a+1+\nu,\dots\right\}\\ f(t),&\quad \nu=0,\ t\in\mathbb{N}_a,\endcases $$ 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.

39A12Discrete version of topics in analysis
26A33Fractional derivatives and integrals (real functions)
34A08Fractional differential equations
39A10Additive difference equations
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