\(\alpha\)-Difference Operator and Its Application on Number Theory

GLOBAL PROPERTIES OF SOLUTIONS OF FOURTH ORDER GENERALIZED alpha-DIFFERENCE EQUATIONS

“…Definition 2.3. [9] The inverse of the Generalized α-difference operator denoted by ∆ −1 α(ℓ) on u(k) is defined as follows. If ∆ α(ℓ) v(k) = u(k), then…”

“…Letting t → ∞ implies that u(k) satisfies (9) for a = ℓ and is recessive. We note that u(k) also satisfies (9) for a = 0. Concerning the monotonicity, we choose any k ∈ [2ℓ, ∞) and any m 3t ≥ k. Then, ∆ ℓ ] (k − ℓ).…”

FINITE SERIES AND COMPLETE SOLUTION OF SECOND ORDER $\alpha$-DIFFERENCE EQUATION

“…If we are able to find a closed form solution of equation (3), which coinciding with the numerical solution of that equation, then we can obtain formula of values of several finite series. In this paper, we extend the theory and applications of ∆ α(ℓ) developed in [12] to generalized second kind α-difference equation…”

“…In 2011, M.Maria Susai Manuel, et.al, [11] have extended the definition of ∆ α to ∆ α(ℓ) defined as ∆ α(ℓ) u(k) = u(k + ℓ)− αu(k) for the real valued function u(k) and ℓ ∈ (0, ∞). In [12], the authors have used the generalized α-difference equation;…”

Oscillation Criteria for Second Order Linear Generalized Difference Equation

“…But recently, when we took up the definition of ∆ as given in (2) we developed the theory of difference equations in a different direction ([8]- [9]). For convenience, we labelled the operator ∆ defined by (2) as ∆ ℓ and by defining its inverse ∆ −1 ℓ , many interesting results and applications in number theory were established (see [8], [10], [11], [12], [13]). …”