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.


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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