izvuzmathИзвестия высших учебных заведений. МатематикаIzvestiya Vysshikh Uchebnykh Zavedenii. Matematika0021-34462076-4626Казанский (Приволжский) федеральный университет10.26907/0021-3446-2026-4-3-24izvuzmath-287Research ArticleСтатьи(s, t)-последовательности Ферма, Ферма–Люка и их приложения к гиперболическим кватернионам(s,t)-Fermat, (s,t)-Fermat–Lucas sequences and their applications to hyperbolic quaternionsАккушХ.AkkuşH.

Хакан Аккуш

г. Эрзинджан

Hakan Akkuş

Erzincan

hakan.akkus@ogr.ebyu.edu.trОзканЭ.ÖzkanE.

Энгин Озкан

г. Стамбул

Engin Özkan

İstanbul

engin.ozkan@marmara.edu.trУниверситет Бинали Йылдырыма (Эрзинджан)ТурцияErzincan Binali Yıldırım UniversityTurkeyУниверситет МармараТурцияMarmara UniversityTurkey202608052026043–24Copyright © Аккуш Х., Озкан Э., 20262026Аккуш Х., Озкан Э.Akkuş H., Özkan E.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://izvuzmath.elpub.ru/jour/article/view/287

Определяются $(s,t)$-последовательности Ферма и Ферма–Люка. Для таких последовательностей приводится множество свойств, таких как характеристическое уравнение и формулы Бине. Получены соотношения между этими последовательностями. Найдены различные представления для нахождения членов этих последовательностей. Показана связь между положительными и отрицательными членами последовательностей и получено соотношение между тремя последовательными членами. Представлены порождающие функции и формулы сумм для этих последовательностей. Кроме того, вычислены специальные тождества для этих последовательностей, такие как тождество Каталана, Мелхэма,   Оканя и другие. Исследованы связи $(s,t)$-последовательности Ферма с последовательностями Фибоначчи, Пелля, Якобсталя, Орезма и с балансирующей последовательностью, а также связи $(s,t)$-последовательности Ферма-Люка   с последовательностями Люка, Пелля–Люка, Якобсталя–Люка, Орезма–Люка и  с балансирующей последовательностью Люка. Более того, предложено применение $(s,t)$-последовательностей Ферма и Ферма–Люка к гиперболическим кватернионам. Определены гиперболические $(s,t)$-кватернионы Ферма и Ферма–Люка. Для этих гиперболических кватернионов приведено множество свойств, таких как формулы Бине. Наконец, члены $(s,t)$-последовательностей Ферма и Ферма–Люка ассоциированы со значениями гиперболических кватернионов.

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.

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The authors declare that there are no conflicts of interest present.