A Note on Generalized <i>k</i>-Order F&L Hybrinomials
In this study, we introduce generalized <i>k</i>-order Fibonacci and Lucas (F&L) polynomials that allow the derivation of well-known polynomial and integer sequences such as the sequences of <i>k</i>-order Pell polynomials, <i>k</i>-order Jacobsthal polynomial...
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2025-01-01
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| author | Süleyman Aydınyüz Gül Karadeniz Gözeri |
| author_facet | Süleyman Aydınyüz Gül Karadeniz Gözeri |
| author_sort | Süleyman Aydınyüz |
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| description | In this study, we introduce generalized <i>k</i>-order Fibonacci and Lucas (F&L) polynomials that allow the derivation of well-known polynomial and integer sequences such as the sequences of <i>k</i>-order Pell polynomials, <i>k</i>-order Jacobsthal polynomials and <i>k</i>-order Jacobsthal F&L numbers. Within the scope of this research, a generalization of hybrid polynomials is given by moving them to the <i>k</i>-order. Hybrid polynomials defined by this generalization are called <i>k</i>-order F&L hybrinomials. A key aspect of our research is the establishment of the recurrence relations for generalized <i>k</i>-order F&L hybrinomials. After we give the recurrence relations for these hybrinomials, we obtain the generating functions of hybrinomials, shedding light on some of their important properties. Finally, we introduce the matrix representations of the generalized <i>k</i>-order F&L hybrinomials and give some properties of the matrix representations. |
| format | Article |
| id | doaj-art-fe1ae0dd32e541718e299b7394c64350 |
| institution | Kabale University |
| issn | 2075-1680 |
| language | English |
| publishDate | 2025-01-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Axioms |
| spelling | doaj-art-fe1ae0dd32e541718e299b7394c643502025-01-24T13:22:14ZengMDPI AGAxioms2075-16802025-01-011414110.3390/axioms14010041A Note on Generalized <i>k</i>-Order F&L HybrinomialsSüleyman Aydınyüz0Gül Karadeniz Gözeri1Department of Mathematics, Pamukkale University, Denizli 20160, TurkeyDepartment of Mathematics, Istanbul University, Istanbul 34320, TurkeyIn this study, we introduce generalized <i>k</i>-order Fibonacci and Lucas (F&L) polynomials that allow the derivation of well-known polynomial and integer sequences such as the sequences of <i>k</i>-order Pell polynomials, <i>k</i>-order Jacobsthal polynomials and <i>k</i>-order Jacobsthal F&L numbers. Within the scope of this research, a generalization of hybrid polynomials is given by moving them to the <i>k</i>-order. Hybrid polynomials defined by this generalization are called <i>k</i>-order F&L hybrinomials. A key aspect of our research is the establishment of the recurrence relations for generalized <i>k</i>-order F&L hybrinomials. After we give the recurrence relations for these hybrinomials, we obtain the generating functions of hybrinomials, shedding light on some of their important properties. Finally, we introduce the matrix representations of the generalized <i>k</i>-order F&L hybrinomials and give some properties of the matrix representations.https://www.mdpi.com/2075-1680/14/1/41Fibonacci polynomialsgeneralized <i>k</i>-order Fibonacci and Lucas polynomialshybrid numbershybrid polynomialsgeneralized <i>k</i>-order Fibonacci and Lucas hybrinomials |
| spellingShingle | Süleyman Aydınyüz Gül Karadeniz Gözeri A Note on Generalized <i>k</i>-Order F&L Hybrinomials Axioms Fibonacci polynomials generalized <i>k</i>-order Fibonacci and Lucas polynomials hybrid numbers hybrid polynomials generalized <i>k</i>-order Fibonacci and Lucas hybrinomials |
| title | A Note on Generalized <i>k</i>-Order F&L Hybrinomials |
| title_full | A Note on Generalized <i>k</i>-Order F&L Hybrinomials |
| title_fullStr | A Note on Generalized <i>k</i>-Order F&L Hybrinomials |
| title_full_unstemmed | A Note on Generalized <i>k</i>-Order F&L Hybrinomials |
| title_short | A Note on Generalized <i>k</i>-Order F&L Hybrinomials |
| title_sort | note on generalized i k i order f l hybrinomials |
| topic | Fibonacci polynomials generalized <i>k</i>-order Fibonacci and Lucas polynomials hybrid numbers hybrid polynomials generalized <i>k</i>-order Fibonacci and Lucas hybrinomials |
| url | https://www.mdpi.com/2075-1680/14/1/41 |
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