A unifying approach to the difference operators and their applications

abstract: In the present paper, we introduce the idea of difference operators ∆ and ∆(α)(α ∈ R) and establish certain results which have several applications in Functional as well as Numerical analysis. Indeed, the operator ∆ generalizes several difference operators defined by Kızmaz [1], Et [2], Et and Çolak [3], Malkowsky and Parashar [4], Et [5], Malkowsky et al. [6], Baliarsingh [7] and many others.


Introduction and definitions
By Γ(p), we denote the Gamma function of a real number p and p / ∈ {0, −1, −2, −3, . . .}.By the definition, it can be expressed as an improper integral i.e.
x k+i , defined by Et and Çolak [2].
Recently, different classes of difference sequences have been introduced and their different properties including topological structures, duals, and matrix transformations have been studied by Tripathy [8], Et and Basarir [9], Dutta and Baliarsingh [10,12], Mursaleen [11], Tripathy et al. [13,15], Asma and Çolak [14] and many others (see [16]- [23]).In this article, we unify most of the difference operators studied by earlier authors and extend their results in a more general and comprehensive way.

Main results
In this section, we state some interesting results concerning the linearity property of the difference operators ∆ α , ∆ (α) , ∆ −α and ∆ (−α) .Also, we discuss certain relations among these operators.
Proof: Proof is trivial, hence omitted.✷ Theorem 2.2.If α and β are two real numbers, then Proof: Proof follows from Theorem 2.1, so we omit the details.✷ Theorem 2.3.If α is a real number, then where Id is the identity operator in w.
Proof: (i) The proof of this theorem is divided into two parts.First we prove the theorem for any positive integer α which can be obtained by using inductive principle.Suppose x ∈ w and for α = 1, we have This shows that ∆ • ∆ −1 ≡ Id in w.By principle of induction one can establish ∆ r • ∆ −r ≡ Id in w.Similarly, for a fraction α, we can show that ∆ α • ∆ −α ≡ Id in w.
(ii) In view of the proof of (i), that of (ii) is similar, so we omit it.✷ Theorem 2.4.For a positive integer α and x ∈ w, Proof: (i) We prove the theorem by induction principle.For α = 1 and x ∈ w, we have This completes the Basis step.Let us assume that the theorem is true for a natural number r, i.e. (∆ r x) k = (−1) r (∆ (r) x) k+r .Now, we take r) x) k+r ), (by the assumption).
This completes the proof.
(ii) The proof is similar to that of (i).✷ Theorem 2.5.For any real α and x ∈ w, we have where (α Proof: The proof is straightforward from the definition, so we omit it.✷ Let x = (x k ) and y = (y k ) be two sequences in w.We define the product of x and y as xy = (x k y k ).Now, the first forward and backward differences of xy are given by ∆(xy) = (x k y k − x k+1 y k+1 ) and ∆ (1) (xy) = (x k y k − x k−1 y k−1 ), respectively.The basic objective of this part is to find the α-th difference of product sequence xy where α is a positive integer.So, we state the following theorems.
Theorem 2.6.(Leibnitz Theorem).Let α = n be a positive integer and x, y ∈ w, then Proof: This proceeds by induction on natural numbers n, the result being trivial for n = 0 and reducing for n = 1 to the well-known rule for differentiating a product (once).Suppose n = 1 and x, y ∈ w, we obtain that Let us assume that the theorem holds for a positive integer r which can be stated as This leads to the completion of the proof.✷

Application to the numerical analysis
In this section, we discuss some numerical applications of the difference operators ∆ α , ∆ (α) , ∆ −α and ∆ (−α) .In fact, these calculations are often used in finding interpolating polynomial, numerical differentiation and integration of a function where we write f = (f k ) as a sequence of functional values of f (x) at x 1 , x 2 , x 3 , . . . .The well known Newton's forward and backward interpolation formula for f (x) can be explained with the help of these operators.Now, we consider some particular cases.