On generalized Fibonacci quaternion sequences with periodic coefficients: Applications

¹ Department of Mathematics, Faculty of Sciences, Univ. My Ismail, Meknés, Morocco
² Instituto de Matemática INMA, Federal University of Mato Grosso do Sul, Campo Grande, MS. Brazil
³ Department of Mathematics, Ankara Hacı Bayram Veli University, Ankara, 06900, Turkey

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Received: 17 February 2026
Accepted: 2 April 2026

Abstract

This paper develops a comprehensive theory of r-generalized Fibonacci quaternion sequences with periodic coefficients, extending and unifying the classical theory of Fibonacci quaternion sequences. We establish explicit combinatorial formulas for both constant and periodic coefficients cases with particular emphasis on the role of periodicity in the sequence structure. Special attention is given to the case r = 2, leading to novel applications for Pell, h-Pell, and Pell-Lucas quaternion sequences. Our results generalize several existing theorems in the literature, while providing new insights into the combinatorial nature of r-generalized Fibonacci quaternion sequences.