Abstract
Abstract In this paper, we introduce generalizations of Quaternacci sequences (Quaternaccis), called Split Quaternacci sequences, which arose on a base of split quaternion algebras. Explicit and recurrent formulae for Split Quaternacci sequences are given, as well as generating functions. Also, matrices related to Split Quaternaccis sequences are investigated. Moreover, new identities connecting Horadam sequences with other known sequences are generated. Analogous identities for Horadam quaternions and split Horadam quaternions are proved.
Highlights
In [1], we introduced new sequences called Quaternaccis
The idea of Quaternaccis arose during research on two-parametric quasi-Fibonacci numbers of 7th and 9th order [2]
From one side nth roots of unity form a cyclic group under multiplication, and from the other some sums of these roots are linearly independent
Summary
In [1], we introduced new sequences called Quaternaccis. The idea of Quaternaccis arose during research on two-parametric quasi-Fibonacci numbers of 7th and 9th order [2] (see [3], where some basic ideas were presented). Two-parametric quasi-Fibonacci numbers of 7th order are members of sequences (An,7(δ, γ)), (Bn,7(δ, γ)) and (Cn,7(δ, γ)) defined by the following relations:. [1, Definition 1] Quaternacci sequences (shortly Quaternaccis) An(b, c, d), Bn(b, c, d), Cn(b, c, d), Dn(b, c, d) are defined by the following relations:. From Definition 1 we obtain an explicit formula for powers of quaternions. During our investigations we obtained a lot of interesting results, on quaternion structure itself [1]. We generate some new identities, called “bridges,” connecting sequences in question with the other known sequences (we give analogous identities for Quaternaccis as well)
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