The Generalized Difference Operator of the 𝐧𝐭𝐡 Kind

In this chapter, the authors extend the theory of the generalized difference Operator ∆L to the generalized difference operator of the 𝒏 𝒕𝒉kind denoted by ∆L Where L =𝑳 = {𝒍𝟏,𝒍𝟐,….𝒍𝒏} of positive reals𝒍𝟏,𝒍𝟐,….𝒍𝒏and obtain some interesting results on the relation between the generalized polynomial factorial of the first kind, 𝒏 𝒕𝒉kind and algebraic polynomials. Also formulae for the sum of the general partial sums of products of several powers of consecutive terms of an Arithmetic progression in number theory are derived.