Some Applications of Higher Order Generalized α− Difference Operator

In this paper, we derive the discrete version of the Bernoulli’s formula according to the generalized αdifference operator for negative `, and to find the sum of several type of arithmetic series in the field of Numerical Methods. Suitable example are provided to illustrate the main results.


1.Introduction
The theory of difference equations is based on the operator ∆ defined as ∆u(k) = u(k + 1) − u(k), k ∈ N(0) = {0, 1, 2, ...}. (1) Eventhough many authors [1], [11], [14] have suggested the definition of ∆ as and no significant progress took place in the field of numerical methods, they took up the definition of ∆ as given in (2), and developed the theory of difference equations in a different direction and many interesting results were obtained in number theory.
For convenience, they labelled the operator ∆ defined by (2) as ∆ and its inverse by ∆ −1 .When ∆ is operated on a complex function u(k) and considering to be real, some new qualitative properties like rotatory, expanding, shrinking, spiral and weblike were noticed.The results obtained can be found in [15].Jerzy Popenda, et al., [18], while discussing the behavior of solutions of a particular type of difference equation, defined ∆ α as ∆ α u(k) = u(k + 1) − αu(k).This definition of ∆ α is being ignored for a long time.In [5], we have generalized the definition of ∆ α by ∆ α( ) defined as ∆ α( ) u(k) = u(k + ) − αu(k) for the real valued function u(k) and ∈ (0, ∞) and also obtained the solutions of certain types of generalized α-difference equations, in particular, the generalized Clairaut's α-difference equation, generalized Euler α-difference equation and the generalized α-Bernoulli polynomial B α(n) (k, ), which is a solution of the α-difference equation u(k + ) − αu(k) = nk n−1 , for n ∈ N(1) [19], [4].
Recently, G.B.A.Xavier, et.al.extended from the definition of generalized α-difference operator of n th kind and to obtain the formula for sum of partial sums of various types of arithmetic-geometric progression in the field of Numerical Analysis [17].
With this background, in this paper we derive the generalized discrete α−Bernoulli's formula and to obtain the formula for sum of several types of arithmetic and geometric series using the stirling numbers of first and second kind respectively.
Throughout this paper, we make use of the following notations:

Preliminaries
In this section, we present some basic definitions and preliminary results which will be useful for further subsequent discussions.
and inverse is defined by where v(j) is constant for all k ∈ N (j).
Lemma 2.4 Let u(k) and v(k) be any two real valued functions.Then, and

Generalized Discrete α− Bernoulli's Formula
In this section, for negative index − , we derive the discrete α− Bernoulli's formula and establish as the sum of general partial sums of products of polynomials and polynomial factorials using the inverse of generalized α-difference operator and stirling numbers of first kind and second kind respectively.
Proof: From Lemma 2.4, we have − ) and applying the limit from + j to k, we obtain Similarly again operating ∆ −1 α( ) on both sides and applying the limit from 2 + j to k and which can be expressed as .
The proof completes by continuing this process.
Corollary 3.2 Let k n be the generalized polynomial then .