Abstract
In this paper, we study two generalizations of dual-hyperbolic balancing numbers: dual-hyperbolic Horadam numbers and dual-hyperbolic k-balancing numbers. We give Catalan’s identity, Cassini’s identity, and d’Ocagne’s identity for them.
Highlights
We focus on the balancing numbers, the Lucas-balancing numbers, and some modifications and generalizations of these numbers
In the same way, using Formulas (ii)–(v) of Theorem 3, one can obtain properties of other classes of dual-hyperbolic numbers defined in this paper
We discussed an extended version of dual-hyperbolic balancing numbers, dual-hyperbolic Horadam numbers, and dual-hyperbolic k-balancing numbers
Summary
([9]) The Binet-type formula for k-Lucas-balancing numbers is: αnk + βnk for n ≥ 0, k ≥ 1, where αk = 3k + 9k2 − 1, β k = 3k − 9k2 − 1. We define and study dual-hyperbolic balancing numbers and some of their generalizations. We define the nth dual-hyperbolic k-balancing number DH Bnk , the nth dual-hyperbolic k-Lucas balancing number DHCnk , the nth dual-hyperbolic k-cobalancing number For k = 1, we obtain the Binet-type formulas for the dual-hyperbolic balancing numbers, dual-hyperbolic Lucas-balancing numbers, etc.
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