Application of the Generalized Difference Operator of the nth Kind

In this paper, the authors extend the theory of the generalized difference operator ∆l to the Generalized difference operator of the n th kind L = {l1, l2, l3,. . . , ln} for the positive reals l1, l2, l3,. . . , ln and obtain some interesting results on n th kind. Also by defining its inverse, we establish a few formulae for the sum of third partial sums of products of several powers of the arithmetic progressions in number theory. Appropriate examples are provided to illustrate the results.