In [1, 2], the authors discussed properties of the generalized falling function, a corresponding power rule for fractional delta-operators and the commutativity of fractional sums.
For example, Atici and Eloe [3] discussed the properties of the generalized falling function, the corresponding power rule for fractional delta operators, and the commutativity of fractional sums.
Definition 2.1 We define the generalized falling function by t α ̲ : = Γ ( t + 1 ) Γ ( t + 1 − α ), for any t and α, for which the right-hand side is defined.
For example, Atici and Eloe discussed the properties of the generalized falling function, a corresponding power rule for fractional delta-operators, and the commutativity of fractional sums in [1].
We define the generalized falling function by (t^{underline{alpha}}:=frac{Gamma (t+1)}{Gamma(t+1-alpha)}), for any t and α for which the right-hand side is defined.
As a first interesting case, let us consider the generalized falling factorial polynomials ((x|lambda)_{n}=x x-lambda cdots(x-(n-1)lambda)), for (ngeq1), and ((x|lambda)_{0}=x x-lambda cdots
A recent interest in discrete fractional calculus has been shown by many scholars, such as Atici and Eloe, who in[15] discuss properties of the generalized falling function, a corresponding power rule for fractional delta-operators and the commutivity of fractional sums.
Averages (note 10), from the arithmetic mean to the weighted generalized average, fall between the results obtained using intersections and unions: Conjunctive (union; logical AND; t-norms; Min): given partial scores, the aggregate score will be high iff the partial scores are high.
In fact most of the generalized commutativity notions fall in the subclass of weak compatibility in the setting of a unique common fixed point (or unique point of coincidence).
In view of the results of Dorić et al. [92] (see also Alghamdi et al. [94], Kadelburg et al. [96], Pant and Bisht [40] and Bisht and Rakocević [95]) most of the generalized commutativity notions fall in the subclass of weak compatibility in the setting of a unique common fixed point (or unique point of coincidence).
