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On the generalized bi-periodic Lucas quaternions

As is well known, the Fibonacci sequence {Fn} is generated from the recurrence relation Fn = Fn−1 + Fn−2 with initial condition F0 = 0, F1 = 1, and the Lucas sequence {Ln} satisfies the same recurrence relation with initial condition L0 = 2, L1 = 1. The Fibonacci and Lucas numbers possess many interesting properties and appear in a variety of application fields [13]. There exist many generalizations of Fibonacci and Lucas sequences in the literature. Two of them are the bi-periodic Fibonacci sequence {un} [7] and the bi-periodic Lucas sequence {vn} [1] respectively defined by