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

The author introduces in the ν th -order fractional sum of f: a as

Δ a -ν f(t):=1 Γ(ν) s=a t-ν t-σ(s)ν-1 f(s),ν>0,t a+ν :=a+ν,a+1+ν,f(t),ν=0,t a ,

and the ν th -order fractional difference as

(Δ a ν f)(t):=Δ N Δ a -(N-ν) f(t),t a+N-ν ,

where ν0 is given and N is such that N-1<νN. 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 ν th -order fractional difference equation

Δ a+ν-N ν y(t)=f(t),t a ,

and the uniqueness of the solution for the latter equation completed with the initial values

Δ i y(a+ν-N)=A i ,i{0,1,,N-1},A i ,

are established. Finally, the developed theory is illustrated by two examples.

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