On hypercomplex numbers with third-order $k$-Jacobsthal numbers
UDC 517.5 We introduce a new family of hypercomplex numbers by using third-order $k$-Jacobsthal numbers. These sequences are called the third-order $k$-Jacobsthal $2^{r}$-ions. We present some algebraic properties of the third order $k$-Jacobsthal $2^{r}$-ions, such as the recurrence relation, the B...
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| Datum: | 2026 |
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| Format: | Artikel |
| Sprache: | Englisch |
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Institute of Mathematics, NAS of Ukraine
2026
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| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/8969 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860513347201400832 |
|---|---|
| author | Morales, Gamaliel Morales, Gamaliel |
| author_facet | Morales, Gamaliel Morales, Gamaliel |
| author_sort | Morales, Gamaliel |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2026-03-21T13:34:29Z |
| description | UDC 517.5
We introduce a new family of hypercomplex numbers by using third-order $k$-Jacobsthal numbers. These sequences are called the third-order $k$-Jacobsthal $2^{r}$-ions. We present some algebraic properties of the third order $k$-Jacobsthal $2^{r}$-ions, such as the recurrence relation, the Binet formula, generating function, exponential generation function, the d'Ocagne identity, and the Cassini identity. Further, we obtain the matrix representation of the third-order $k$-Jacobsthal $2^{r}$-ions. |
| doi_str_mv | 10.3842/umzh.v77i10.8969 |
| first_indexed | 2026-03-24T03:43:14Z |
| format | Article |
| fulltext | |
| id | umjimathkievua-article-8969 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T03:43:14Z |
| publishDate | 2026 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | |
| spelling | umjimathkievua-article-89692026-03-21T13:34:29Z On hypercomplex numbers with third-order $k$-Jacobsthal numbers On hypercomplex numbers with third-order $k$-Jacobsthal numbers Morales, Gamaliel Morales, Gamaliel Binet formula hyper complex number recurrence relation third-order Jacobsthal number UDC 517.5 We introduce a new family of hypercomplex numbers by using third-order $k$-Jacobsthal numbers. These sequences are called the third-order $k$-Jacobsthal $2^{r}$-ions. We present some algebraic properties of the third order $k$-Jacobsthal $2^{r}$-ions, such as the recurrence relation, the Binet formula, generating function, exponential generation function, the d'Ocagne identity, and the Cassini identity. Further, we obtain the matrix representation of the third-order $k$-Jacobsthal $2^{r}$-ions. УДК 517.5 Про гіперкомплексні числа з $k$-числами Якобсталя третього порядку Уведено нову родину гіперкомплексних чисел, побудовану за допомогою $k$-чисел Якобсталя третього порядку. Ці послідовності названо $2^{r}$-іонами $k$-чисел Якобсталя третього порядку. Наведено деякі алгебраїчні властивості $2^{r}$-іонів $k$-чисел Якобсталя третього порядку, зокрема рекурентне співвідношення, формулу Біне, твірну функцію, експоненціальну твірну функцію, тотожність д’Оканя та тотожність Кассіні. Отримано також матричне подання $2^{r}$-іонів $k$-чисел Якобсталя третього порядку. Institute of Mathematics, NAS of Ukraine 2026-03-21 Article Article https://umj.imath.kiev.ua/index.php/umj/article/view/8969 10.3842/umzh.v77i10.8969 Ukrains’kyi Matematychnyi Zhurnal; Vol. 77 No. 10 (2025); 644 Український математичний журнал; Том 77 № 10 (2025); 644 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/8969/10570 Copyright (c) 2025 Gamaliel Morales |
| spellingShingle | Morales, Gamaliel Morales, Gamaliel On hypercomplex numbers with third-order $k$-Jacobsthal numbers |
| title | On hypercomplex numbers with third-order $k$-Jacobsthal numbers |
| title_alt | On hypercomplex numbers with third-order $k$-Jacobsthal numbers |
| title_full | On hypercomplex numbers with third-order $k$-Jacobsthal numbers |
| title_fullStr | On hypercomplex numbers with third-order $k$-Jacobsthal numbers |
| title_full_unstemmed | On hypercomplex numbers with third-order $k$-Jacobsthal numbers |
| title_short | On hypercomplex numbers with third-order $k$-Jacobsthal numbers |
| title_sort | on hypercomplex numbers with third-order $k$-jacobsthal numbers |
| topic_facet | Binet formula hyper complex number recurrence relation third-order Jacobsthal number |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/8969 |
| work_keys_str_mv | AT moralesgamaliel onhypercomplexnumberswiththirdorderkjacobsthalnumbers AT moralesgamaliel onhypercomplexnumberswiththirdorderkjacobsthalnumbers |