In the present paper, for a positive integer r, we study bi-periodic r-Fibonacci sequence and its family of companion sequences, bi-periodic r-Lucas sequence of type s with 1 ≤ s ≤ r, which extend the classical Fibonacci and Lucas sequences. Afterwards, we establish the link between the bi-periodic r-Fibonacci sequence and its companion sequences. Furthermore, we give their properties as linear recurrence relations, generating functions, explicit formulas and Binet forms.
This paper deals with developing a new class of quaternions, called hyper-dual Horadam quaternions which are constructed from the quaternions whose components are hyper-dual Horadam numbers. We investigate the algebraic properties of these quaternions.
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