Akkuş H., Özkan E.
RUSSIAN MATHEMATICS, sa.4, ss.3-24, 2026 (ESCI, Scopus)
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Yayın Türü:
Makale / Tam Makale
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Basım Tarihi:
2026
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Doi Numarası:
10.26907/0021-3446-2026-4-3-24
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Dergi Adı:
RUSSIAN MATHEMATICS
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Derginin Tarandığı İndeksler:
Scopus, Emerging Sources Citation Index (ESCI), MathSciNet, zbMATH
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Sayfa Sayıları:
ss.3-24
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Marmara Üniversitesi Adresli:
Evet
Özet
In this study, we define the $\left(s,t\right)$-Fermat and $\left(s,t\right)$-Fermat–Lucas sequences. For these sequences, we give many features such as the characteristic equation and Binet formulas. We obtain the relations between these sequences. Various representations have been found for determining the terms of these sequences. We show the relationship between the positive and negative terms of the sequences and get the relation between three consecutive terms. Generating functions and summation formulas for these sequences are presented. In addition, special identities for these sequences are derived, such as the Catalan, Melham, and D'Ocagne identities, among others. We examine the relationships of the $(s,t)$-Fermat sequence with the Fibonacci, Pell, Jacobsthal, Balancing, Oresme sequences, as well as the relationships of the $(s,t)$-Fermat–Lucas sequence with the Lucas, Pell–Lucas, Jacobsthal–Lucas, Balancing–Lucas, Oresme–Lucas sequences. Moreover, an application of the $(s,t)$-Fermat and $(s,t)$-Fermat–Lucas sequences to hyperbolic quaternions is proposed. We define hyperbolic $(s,t)$-Fermat and $(s,t)$-Fermat–Lucas quaternions. For these hyperbolic quaternions, numerous properties are established, such as Binet formulas. Finally, the terms of the $(s,t)$-Fermat and $(s,t)$-Fermat-Lucas sequences are associated with their hyperbolic quaternion values.