$q$-Numbers of quantum groups, Fibonacci numbers, and orthogonal polynomials
We obtain algebraic relations (identities) for $q$-numbers that do not contain $q^{α}$-factors. We derive a formula that expresses any $q$-number $[x]$ in terms of the $q$-number [2]. We establish the relationship between the $q$-numbers $[n]$ and the Fibonacci numbers, Chebyshev polynomials, and ot...
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| Date: | 1998 |
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| Main Authors: | , |
| Format: | Article |
| Language: | Ukrainian English |
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Institute of Mathematics, NAS of Ukraine
1998
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/4854 |
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| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860511034286014464 |
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| author | Kachurik, I. I. Качурик, І. І. |
| author_facet | Kachurik, I. I. Качурик, І. І. |
| author_sort | Kachurik, I. I. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T21:15:53Z |
| description | We obtain algebraic relations (identities) for $q$-numbers that do not contain $q^{α}$-factors. We derive a formula that expresses any $q$-number $[x]$ in terms of the $q$-number [2]. We establish the relationship between the $q$-numbers $[n]$ and the Fibonacci numbers, Chebyshev polynomials, and other special functions. The sums of combinations of $q$-numbers, in particular, the sums of their powers, are calculated. Linear and bilinear generating functions are found for “natural” $q$-numbers. |
| first_indexed | 2026-03-24T03:06:28Z |
| format | Article |
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| id | umjimathkievua-article-4854 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T03:06:28Z |
| publishDate | 1998 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/18/eea237b4d9082117f87092c63a209518.pdf |
| spelling | umjimathkievua-article-48542020-03-18T21:15:53Z $q$-Numbers of quantum groups, Fibonacci numbers, and orthogonal polynomials $q$-Числа квантових груп, числа Фібоначчі і ортогональні многочлени Kachurik, I. I. Качурик, І. І. We obtain algebraic relations (identities) for $q$-numbers that do not contain $q^{α}$-factors. We derive a formula that expresses any $q$-number $[x]$ in terms of the $q$-number [2]. We establish the relationship between the $q$-numbers $[n]$ and the Fibonacci numbers, Chebyshev polynomials, and other special functions. The sums of combinations of $q$-numbers, in particular, the sums of their powers, are calculated. Linear and bilinear generating functions are found for “natural” $q$-numbers. Одержано алгебраїчні співвідношення (тотожності) між $q$-числами, які не містять $α$-множників. Виведено формулу, яка виражає будь-яке $q$- число $[x]$ через $q$-число [2]. Встановлено зв'язок $q$ - чисел $[n]$ з числами Фібоначчі, многочленами Чебишева та з іншими спеціальними функціями. Обчислено суми комбінацій $q$- чисел, зокрема суми їх степенів. Знайдено лінійні та біліпійні породжуючі функції для „натуральних" $q$-чисел. Institute of Mathematics, NAS of Ukraine 1998-08-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/4854 Ukrains’kyi Matematychnyi Zhurnal; Vol. 50 No. 8 (1998); 1055-1063 Український математичний журнал; Том 50 № 8 (1998); 1055-1063 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/4854/6407 https://umj.imath.kiev.ua/index.php/umj/article/view/4854/6408 Copyright (c) 1998 Kachurik I. I. |
| spellingShingle | Kachurik, I. I. Качурик, І. І. $q$-Numbers of quantum groups, Fibonacci numbers, and orthogonal polynomials |
| title | $q$-Numbers of quantum groups, Fibonacci numbers, and orthogonal polynomials |
| title_alt | $q$-Числа квантових груп, числа Фібоначчі і ортогональні многочлени |
| title_full | $q$-Numbers of quantum groups, Fibonacci numbers, and orthogonal polynomials |
| title_fullStr | $q$-Numbers of quantum groups, Fibonacci numbers, and orthogonal polynomials |
| title_full_unstemmed | $q$-Numbers of quantum groups, Fibonacci numbers, and orthogonal polynomials |
| title_short | $q$-Numbers of quantum groups, Fibonacci numbers, and orthogonal polynomials |
| title_sort | $q$-numbers of quantum groups, fibonacci numbers, and orthogonal polynomials |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/4854 |
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