New Identities and Equation Solutions Involving k-Oresme and k-Oresme–Lucas Sequences
Number sequences are among the research areas of interest in both number theory and linear algebra. In particular, the study of matrix representations of recursive sequences is important in revealing the structural properties of these sequences. In this study, the relationships between the elements...
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| author | Bahar Demirtürk |
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| description | Number sequences are among the research areas of interest in both number theory and linear algebra. In particular, the study of matrix representations of recursive sequences is important in revealing the structural properties of these sequences. In this study, the relationships between the elements of the k-Fibonacci and k-Oresme sequences were analyzed using matrix algebra through matrix structures created by connecting the characteristic equations and roots of these sequences. In this context, using the properties of these matrices, the identities <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>A</mi><mi>n</mi><mn>2</mn></msubsup><mo>−</mo><msub><mi>A</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><msub><mi>A</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>=</mo><msup><mi>k</mi><mrow><mo>−</mo><mn>2</mn><mi>n</mi></mrow></msup></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>A</mi><mi>n</mi><mn>2</mn></msubsup><mo>−</mo><msub><mi>A</mi><mi>n</mi></msub><msub><mi>A</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>+</mo><mstyle scriptlevel="0" displaystyle="true"><mfrac><mn>1</mn><mrow><msup><mi>k</mi><mn>2</mn></msup></mrow></mfrac></mstyle><msubsup><mi>A</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow><mn>2</mn></msubsup><mo>=</mo><msup><mi>k</mi><mrow><mo>−</mo><mn>2</mn><mi>n</mi></mrow></msup></mrow></semantics></math></inline-formula>, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>B</mi><mi>n</mi><mn>2</mn></msubsup><mo>−</mo><msub><mi>B</mi><mi>n</mi></msub><msub><mi>B</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>+</mo><mstyle scriptlevel="0" displaystyle="true"><mfrac><mn>1</mn><mrow><msup><mi>k</mi><mn>2</mn></msup></mrow></mfrac></mstyle><msubsup><mi>B</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow><mn>2</mn></msubsup><mo>=</mo><mo>−</mo><mo stretchy="false">(</mo><msup><mi>k</mi><mn>2</mn></msup><mo>−</mo><mn>4</mn><mo stretchy="false">)</mo><msup><mi>k</mi><mrow><mo>−</mo><mn>2</mn><mi>n</mi></mrow></msup></mrow></semantics></math></inline-formula>, and some generalizations such as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>B</mi><mrow><mrow><mi>n</mi><mo>+</mo><mi>m</mi></mrow></mrow><mn>2</mn></msubsup><mo>−</mo><mo stretchy="false">(</mo><msup><mi>k</mi><mn>2</mn></msup><mo>−</mo><mn>4</mn><mo stretchy="false">)</mo><msub><mi>A</mi><mrow><mi>n</mi><mo>−</mo><mi>t</mi></mrow></msub><msub><mi>B</mi><mrow><mi>n</mi><mo>+</mo><mi>m</mi></mrow></msub><msub><mi>A</mi><mrow><mi>t</mi><mo>+</mo><mi>m</mi></mrow></msub><mo>−</mo><mo stretchy="false">(</mo><msup><mi>k</mi><mn>2</mn></msup><mo>−</mo><mn>4</mn><mo stretchy="false">)</mo><msup><mi>k</mi><mrow><mn>2</mn><mi>t</mi><mo>−</mo><mn>2</mn><mi>n</mi></mrow></msup><msubsup><mi>A</mi><mrow><mi>t</mi><mo>+</mo><mi>m</mi></mrow><mn>2</mn></msubsup><mo>=</mo><msup><mi>k</mi><mrow><mo>−</mo><mn>2</mn><mi>m</mi><mo>−</mo><mn>2</mn><mi>t</mi></mrow></msup><msubsup><mi>B</mi><mrow><mi>n</mi><mo>−</mo><mi>t</mi></mrow><mn>2</mn></msubsup></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>A</mi><mrow><mi>t</mi><mo>+</mo><mi>m</mi></mrow><mn>2</mn></msubsup><mo>−</mo><msub><mi>B</mi><mrow><mi>t</mi><mo>−</mo><mi>n</mi></mrow></msub><msub><mi>A</mi><mrow><mi>n</mi><mo>+</mo><mi>m</mi></mrow></msub><msub><mi>A</mi><mrow><mi>t</mi><mo>+</mo><mi>m</mi></mrow></msub><mo>+</mo><msup><mi>k</mi><mrow><mn>2</mn><mi>n</mi><mo>−</mo><mn>2</mn><mi>t</mi></mrow></msup><msubsup><mi>A</mi><mrow><mi>n</mi><mo>+</mo><mi>m</mi></mrow><mn>2</mn></msubsup><mo>=</mo><msup><mi>k</mi><mrow><mo>−</mo><mn>2</mn><mi>n</mi><mo>−</mo><mn>2</mn><mi>m</mi></mrow></msup><msubsup><mi>A</mi><mrow><mi>t</mi><mo>−</mo><mi>n</mi></mrow><mn>2</mn></msubsup></mrow></semantics></math></inline-formula>, and more were derived, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>m</mi><mo>,</mo><mi>n</mi><mo>,</mo><mi>t</mi><mo>∈</mo><mo>ℤ</mo></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>t</mi><mo>≠</mo><mi>n</mi></mrow></semantics></math></inline-formula>. In addition to this, the solution pairs of the algebraic equations <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><msub><mi>B</mi><mi>p</mi></msub><mi>x</mi><mi>y</mi><mo>+</mo><msup><mi>k</mi><mrow><mo>−</mo><mn>2</mn><mi>p</mi></mrow></msup><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><msup><mi>k</mi><mrow><mo>−</mo><mn>2</mn><mi>q</mi></mrow></msup><msubsup><mi>A</mi><mi>p</mi><mn>2</mn></msubsup></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mo stretchy="false">(</mo><msup><mi>k</mi><mn>2</mn></msup><mo>−</mo><mn>4</mn><mo stretchy="false">)</mo><msub><mi>A</mi><mi>p</mi></msub><mi>x</mi><mi>y</mi><mo>−</mo><mo stretchy="false">(</mo><msup><mi>k</mi><mn>2</mn></msup><mo>−</mo><mn>4</mn><mo stretchy="false">)</mo><msup><mi>k</mi><mrow><mo>−</mo><mn>2</mn><mi>p</mi></mrow></msup><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><msup><mi>k</mi><mrow><mo>−</mo><mn>2</mn><mi>q</mi></mrow></msup><msubsup><mi>B</mi><mi>p</mi><mn>2</mn></msubsup></mrow></semantics></math></inline-formula>, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><msub><mi>B</mi><mi>p</mi></msub><mi>x</mi><mi>y</mi><mo>+</mo><msup><mi>k</mi><mrow><mo>−</mo><mn>2</mn><mi>p</mi></mrow></msup><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><mo>−</mo><mo stretchy="false">(</mo><msup><mi>k</mi><mn>2</mn></msup><mo>−</mo><mn>4</mn><mo stretchy="false">)</mo><msup><mi>k</mi><mrow><mo>−</mo><mn>2</mn><mi>q</mi></mrow></msup><msubsup><mi>A</mi><mi>p</mi><mn>2</mn></msubsup></mrow></semantics></math></inline-formula> are presented, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>A</mi><mi>p</mi></msub></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>B</mi><mi>p</mi></msub></mrow></semantics></math></inline-formula> are k-Oresme and k-Oresme–Lucas numbers, respectively. |
| format | Article |
| id | doaj-art-210d00c539fc4d1e910702fbb528b941 |
| institution | Kabale University |
| issn | 2227-7390 |
| language | English |
| publishDate | 2025-07-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj-art-210d00c539fc4d1e910702fbb528b9412025-08-20T03:58:27ZengMDPI AGMathematics2227-73902025-07-011314232110.3390/math13142321New Identities and Equation Solutions Involving k-Oresme and k-Oresme–Lucas SequencesBahar Demirtürk0Department of Fundamental Sciences, Engineering and Architecture Faculty, Izmir Bakırçay University, 35665 Izmir, TürkiyeNumber sequences are among the research areas of interest in both number theory and linear algebra. In particular, the study of matrix representations of recursive sequences is important in revealing the structural properties of these sequences. In this study, the relationships between the elements of the k-Fibonacci and k-Oresme sequences were analyzed using matrix algebra through matrix structures created by connecting the characteristic equations and roots of these sequences. In this context, using the properties of these matrices, the identities <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>A</mi><mi>n</mi><mn>2</mn></msubsup><mo>−</mo><msub><mi>A</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><msub><mi>A</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>=</mo><msup><mi>k</mi><mrow><mo>−</mo><mn>2</mn><mi>n</mi></mrow></msup></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>A</mi><mi>n</mi><mn>2</mn></msubsup><mo>−</mo><msub><mi>A</mi><mi>n</mi></msub><msub><mi>A</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>+</mo><mstyle scriptlevel="0" displaystyle="true"><mfrac><mn>1</mn><mrow><msup><mi>k</mi><mn>2</mn></msup></mrow></mfrac></mstyle><msubsup><mi>A</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow><mn>2</mn></msubsup><mo>=</mo><msup><mi>k</mi><mrow><mo>−</mo><mn>2</mn><mi>n</mi></mrow></msup></mrow></semantics></math></inline-formula>, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>B</mi><mi>n</mi><mn>2</mn></msubsup><mo>−</mo><msub><mi>B</mi><mi>n</mi></msub><msub><mi>B</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>+</mo><mstyle scriptlevel="0" displaystyle="true"><mfrac><mn>1</mn><mrow><msup><mi>k</mi><mn>2</mn></msup></mrow></mfrac></mstyle><msubsup><mi>B</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow><mn>2</mn></msubsup><mo>=</mo><mo>−</mo><mo stretchy="false">(</mo><msup><mi>k</mi><mn>2</mn></msup><mo>−</mo><mn>4</mn><mo stretchy="false">)</mo><msup><mi>k</mi><mrow><mo>−</mo><mn>2</mn><mi>n</mi></mrow></msup></mrow></semantics></math></inline-formula>, and some generalizations such as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>B</mi><mrow><mrow><mi>n</mi><mo>+</mo><mi>m</mi></mrow></mrow><mn>2</mn></msubsup><mo>−</mo><mo stretchy="false">(</mo><msup><mi>k</mi><mn>2</mn></msup><mo>−</mo><mn>4</mn><mo stretchy="false">)</mo><msub><mi>A</mi><mrow><mi>n</mi><mo>−</mo><mi>t</mi></mrow></msub><msub><mi>B</mi><mrow><mi>n</mi><mo>+</mo><mi>m</mi></mrow></msub><msub><mi>A</mi><mrow><mi>t</mi><mo>+</mo><mi>m</mi></mrow></msub><mo>−</mo><mo stretchy="false">(</mo><msup><mi>k</mi><mn>2</mn></msup><mo>−</mo><mn>4</mn><mo stretchy="false">)</mo><msup><mi>k</mi><mrow><mn>2</mn><mi>t</mi><mo>−</mo><mn>2</mn><mi>n</mi></mrow></msup><msubsup><mi>A</mi><mrow><mi>t</mi><mo>+</mo><mi>m</mi></mrow><mn>2</mn></msubsup><mo>=</mo><msup><mi>k</mi><mrow><mo>−</mo><mn>2</mn><mi>m</mi><mo>−</mo><mn>2</mn><mi>t</mi></mrow></msup><msubsup><mi>B</mi><mrow><mi>n</mi><mo>−</mo><mi>t</mi></mrow><mn>2</mn></msubsup></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>A</mi><mrow><mi>t</mi><mo>+</mo><mi>m</mi></mrow><mn>2</mn></msubsup><mo>−</mo><msub><mi>B</mi><mrow><mi>t</mi><mo>−</mo><mi>n</mi></mrow></msub><msub><mi>A</mi><mrow><mi>n</mi><mo>+</mo><mi>m</mi></mrow></msub><msub><mi>A</mi><mrow><mi>t</mi><mo>+</mo><mi>m</mi></mrow></msub><mo>+</mo><msup><mi>k</mi><mrow><mn>2</mn><mi>n</mi><mo>−</mo><mn>2</mn><mi>t</mi></mrow></msup><msubsup><mi>A</mi><mrow><mi>n</mi><mo>+</mo><mi>m</mi></mrow><mn>2</mn></msubsup><mo>=</mo><msup><mi>k</mi><mrow><mo>−</mo><mn>2</mn><mi>n</mi><mo>−</mo><mn>2</mn><mi>m</mi></mrow></msup><msubsup><mi>A</mi><mrow><mi>t</mi><mo>−</mo><mi>n</mi></mrow><mn>2</mn></msubsup></mrow></semantics></math></inline-formula>, and more were derived, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>m</mi><mo>,</mo><mi>n</mi><mo>,</mo><mi>t</mi><mo>∈</mo><mo>ℤ</mo></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>t</mi><mo>≠</mo><mi>n</mi></mrow></semantics></math></inline-formula>. In addition to this, the solution pairs of the algebraic equations <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><msub><mi>B</mi><mi>p</mi></msub><mi>x</mi><mi>y</mi><mo>+</mo><msup><mi>k</mi><mrow><mo>−</mo><mn>2</mn><mi>p</mi></mrow></msup><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><msup><mi>k</mi><mrow><mo>−</mo><mn>2</mn><mi>q</mi></mrow></msup><msubsup><mi>A</mi><mi>p</mi><mn>2</mn></msubsup></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mo stretchy="false">(</mo><msup><mi>k</mi><mn>2</mn></msup><mo>−</mo><mn>4</mn><mo stretchy="false">)</mo><msub><mi>A</mi><mi>p</mi></msub><mi>x</mi><mi>y</mi><mo>−</mo><mo stretchy="false">(</mo><msup><mi>k</mi><mn>2</mn></msup><mo>−</mo><mn>4</mn><mo stretchy="false">)</mo><msup><mi>k</mi><mrow><mo>−</mo><mn>2</mn><mi>p</mi></mrow></msup><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><msup><mi>k</mi><mrow><mo>−</mo><mn>2</mn><mi>q</mi></mrow></msup><msubsup><mi>B</mi><mi>p</mi><mn>2</mn></msubsup></mrow></semantics></math></inline-formula>, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><msub><mi>B</mi><mi>p</mi></msub><mi>x</mi><mi>y</mi><mo>+</mo><msup><mi>k</mi><mrow><mo>−</mo><mn>2</mn><mi>p</mi></mrow></msup><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><mo>−</mo><mo stretchy="false">(</mo><msup><mi>k</mi><mn>2</mn></msup><mo>−</mo><mn>4</mn><mo stretchy="false">)</mo><msup><mi>k</mi><mrow><mo>−</mo><mn>2</mn><mi>q</mi></mrow></msup><msubsup><mi>A</mi><mi>p</mi><mn>2</mn></msubsup></mrow></semantics></math></inline-formula> are presented, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>A</mi><mi>p</mi></msub></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>B</mi><mi>p</mi></msub></mrow></semantics></math></inline-formula> are k-Oresme and k-Oresme–Lucas numbers, respectively.https://www.mdpi.com/2227-7390/13/14/2321generalized Fibonacci sequencesk-Oresme sequencesmatrix representationsolutions of algebraic equations |
| spellingShingle | Bahar Demirtürk New Identities and Equation Solutions Involving k-Oresme and k-Oresme–Lucas Sequences Mathematics generalized Fibonacci sequences k-Oresme sequences matrix representation solutions of algebraic equations |
| title | New Identities and Equation Solutions Involving k-Oresme and k-Oresme–Lucas Sequences |
| title_full | New Identities and Equation Solutions Involving k-Oresme and k-Oresme–Lucas Sequences |
| title_fullStr | New Identities and Equation Solutions Involving k-Oresme and k-Oresme–Lucas Sequences |
| title_full_unstemmed | New Identities and Equation Solutions Involving k-Oresme and k-Oresme–Lucas Sequences |
| title_short | New Identities and Equation Solutions Involving k-Oresme and k-Oresme–Lucas Sequences |
| title_sort | new identities and equation solutions involving k oresme and k oresme lucas sequences |
| topic | generalized Fibonacci sequences k-Oresme sequences matrix representation solutions of algebraic equations |
| url | https://www.mdpi.com/2227-7390/13/14/2321 |
| work_keys_str_mv | AT bahardemirturk newidentitiesandequationsolutionsinvolvingkoresmeandkoresmelucassequences |