In this paper, we introduce a new family of hypercomplex numbers, called higher-order Leonardo \(2^{s}\) -ions, constructed using higher-order Leonardo numbers. This unified definition aims to encompass several previously known concepts, including ordinary Leonardo numbers, higher-order Leonardo numbers, Leonardo complex numbers, Leonardo quaternions, Leonardo octonions, Leonardo sedenions, and higher-order Leonardo quaternions, within a single comprehensive formula. We derive several formulas and identities for these numbers, such as Binet formula, recurrence relation, ordinary and exponential generating functions, and Vajda’s identity, along with its special cases, namely Catalan’s identity, Cassini’s identity, and d’Ocagne’s identity.
