On Summation Properties and Cassini-like Identity for Generalized k - Lucas Numbers

Abstract

The generalized k - Lucas numbers represent a significant extension of the Lucas numbers and exhibit a strong structural relationship with the k - Fibonacci sequence. This research presents an analytical study of the fundamental algebraic properties of the generalized k - Lucas numbers, which characterized by a distinctive block-partitioned structure. The primary objective of this study is to introduce the theorem of Cassini's identity for the generalized k - Lucas numbers, illustrating the underlying relationships between various terms. Furthermore, this paper has also proven the block and cumulative summation formulas. By establishing the sum of k consecutive terms within any arbitrary block  leads to the extension of these results into a cumulative summation formula from the initial term up to any N^th term. The obtained results can be expressed in a closed form strictly in terms of the values of the Lucas numbers at indices m-1 , m and m+1.