Abstract
In this work we study the generalized Oresme numbers using the algebraic properties of the Fibonacci fundamental system associated with this sequence. We explicitly describe the closed connection between the sequences of the Fibonacci fundamental system and the generalized Oresme numbers. In addition, the matrix approach is considered for studying the combinatorial identities and the generalized Cassini identity for the generalized Oresme numbers. Finally, the analytic representation is established, and the combinatorial and analytical generalized Cassini identity is given. Keywords: Generalized sequence of numbers, Fibonacci fundamental system, combinatorial identities, the Cassini identity
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