k-Mersenne and k-Mersenne-Lucas sedenions.

k-Mersenne and k-Mersenne-Lucas sedenions are specialised extensions of sedenions, a sixteen-dimensional algebraic structure. These variations introduce specific algebraic rules and properties derived from their connection to Mersenne and Lucas numbers. k-Mersenne sedenions are defined by their relationship to k-Mersenne numbers, while k-Mersenne-Lucas sedenions are associated with k-Mersenne-Lucas numbers. In this article firstly k-Mersenne and k-Mersenne-Lucas sedenions are defined. Then the algebraic properties of these sedenions such as norm, conjugate and inner product are examined. The Mersenne and Mersenne-Lucas recurrence relations, Binet's formulas, generating functions and finite sum formulas for these sedenions are derived. These sedenions also reveal interesting connections with established number theory identities such as Catalan's, Cassini's, D'Ocagne's and Vajda's identities, providing further depth to their significance within the mathematical theory and their potential applications across various scientific domains. [ABSTRACT FROM AUTHOR]