Efforts have been made to extend Cassini's identity (also known as Simson's identity) to the k-step or k-bonacci numbers for decades. These efforts have lacked both completeness of result and simplicity of proof, and this question remains open and relevant. In this note, we offer a definitive solution as well as the generalization of both Catalan's and Vajda's identities.

翻译：长期以来，人们一直致力于将Cassini恒等式（亦称Simson恒等式）推广至k步或k-bonacci数。这些尝试在结果的完备性和证明的简洁性方面均有所欠缺，使得该问题至今仍悬而未决且具有研究价值。本文不仅提供了该问题的确凿解答，还同时给出了Catalan恒等式与Vajda恒等式的推广形式。