On Homogeneous Combinations of Linear Recurrence Sequences
Let <inline-formula><math display="inline"><semantics><msub><mrow><mo>(</mo><msub><mi>F</mi><mi>n</mi></msub><mo>)</mo></mrow><mrow><mi>n</mi><mo>≥</mo><mn>0&...
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| Format: | Article |
| Language: | English |
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MDPI AG
2020-12-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/8/12/2152 |
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| author | Marie Hubálovská Štěpán Hubálovský Eva Trojovská |
| author_facet | Marie Hubálovská Štěpán Hubálovský Eva Trojovská |
| author_sort | Marie Hubálovská |
| collection | DOAJ |
| description | Let <inline-formula><math display="inline"><semantics><msub><mrow><mo>(</mo><msub><mi>F</mi><mi>n</mi></msub><mo>)</mo></mrow><mrow><mi>n</mi><mo>≥</mo><mn>0</mn></mrow></msub></semantics></math></inline-formula> be the Fibonacci sequence given by <inline-formula><math display="inline"><semantics><mrow><msub><mi>F</mi><mrow><mi>n</mi><mo>+</mo><mn>2</mn></mrow></msub><mo>=</mo><msub><mi>F</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>+</mo><msub><mi>F</mi><mi>n</mi></msub></mrow></semantics></math></inline-formula>, for <inline-formula><math display="inline"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>0</mn></mrow></semantics></math></inline-formula>, where <inline-formula><math display="inline"><semantics><mrow><msub><mi>F</mi><mn>0</mn></msub><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula> and <inline-formula><math display="inline"><semantics><mrow><msub><mi>F</mi><mn>1</mn></msub><mo>=</mo><mn>1</mn></mrow></semantics></math></inline-formula>. There are several interesting identities involving this sequence such as <inline-formula><math display="inline"><semantics><mrow><msubsup><mi>F</mi><mi>n</mi><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>F</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><mn>2</mn></msubsup><mo>=</mo><msub><mi>F</mi><mrow><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow></semantics></math></inline-formula>, for all <inline-formula><math display="inline"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>0</mn></mrow></semantics></math></inline-formula>. In 2012, Chaves, Marques and Togbé proved that if <inline-formula><math display="inline"><semantics><msub><mrow><mo>(</mo><msub><mi>G</mi><mi>m</mi></msub><mo>)</mo></mrow><mi>m</mi></msub></semantics></math></inline-formula> is a linear recurrence sequence (under weak assumptions) and <inline-formula><math display="inline"><semantics><mrow><msubsup><mi>G</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><mi>s</mi></msubsup><mo>+</mo><mo>⋯</mo><mo>+</mo><msubsup><mi>G</mi><mrow><mi>n</mi><mo>+</mo><mo>ℓ</mo></mrow><mi>s</mi></msubsup><mo>∈</mo><msub><mrow><mo>(</mo><msub><mi>G</mi><mi>m</mi></msub><mo>)</mo></mrow><mi>m</mi></msub></mrow></semantics></math></inline-formula>, for infinitely many positive integers <i>n</i>, then <i>s</i> is bounded by an effectively computable constant depending only on <i>ł</i> and the parameters of <inline-formula><math display="inline"><semantics><msub><mrow><mo>(</mo><msub><mi>G</mi><mi>m</mi></msub><mo>)</mo></mrow><mi>m</mi></msub></semantics></math></inline-formula>. In this paper, we shall prove that if <inline-formula><math display="inline"><semantics><mrow><mi>P</mi><mo>(</mo><msub><mi>x</mi><mn>1</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>x</mi><mo>ℓ</mo></msub><mo>)</mo></mrow></semantics></math></inline-formula> is an integer homogeneous <i>s</i>-degree polynomial (under weak hypotheses) and if <inline-formula><math display="inline"><semantics><mrow><mi>P</mi><mrow><mo>(</mo><msub><mi>G</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>G</mi><mrow><mi>n</mi><mo>+</mo><mo>ℓ</mo></mrow></msub><mo>)</mo></mrow><mo>∈</mo><msub><mrow><mo>(</mo><msub><mi>G</mi><mi>m</mi></msub><mo>)</mo></mrow><mi>m</mi></msub></mrow></semantics></math></inline-formula> for infinitely many positive integers <i>n</i>, then <i>s</i> is bounded by an effectively computable constant depending only on <i>ℓ</i>, the parameters of <inline-formula><math display="inline"><semantics><msub><mrow><mo>(</mo><msub><mi>G</mi><mi>m</mi></msub><mo>)</mo></mrow><mi>m</mi></msub></semantics></math></inline-formula> and the coefficients of <i>P</i>. |
| first_indexed | 2024-03-10T14:21:25Z |
| format | Article |
| id | doaj.art-73829ccb11984461ae7aa0ce38d0b625 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T14:21:25Z |
| publishDate | 2020-12-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-73829ccb11984461ae7aa0ce38d0b6252023-11-20T23:19:56ZengMDPI AGMathematics2227-73902020-12-01812215210.3390/math8122152On Homogeneous Combinations of Linear Recurrence SequencesMarie Hubálovská0Štěpán Hubálovský1Eva Trojovská2Department of Technical Education, Faculty of Education, University of Hradec Králové, 500 03 Hradec Králové, Czech RepublicDepartment of Applied Cybernetics, Faculty of Science, University of Hradec Králové, 500 03 Hradec Králové, Czech RepublicDepartment of Mathematics, Faculty of Science, University of Hradec Králové, 500 03 Hradec Králové, Czech RepublicLet <inline-formula><math display="inline"><semantics><msub><mrow><mo>(</mo><msub><mi>F</mi><mi>n</mi></msub><mo>)</mo></mrow><mrow><mi>n</mi><mo>≥</mo><mn>0</mn></mrow></msub></semantics></math></inline-formula> be the Fibonacci sequence given by <inline-formula><math display="inline"><semantics><mrow><msub><mi>F</mi><mrow><mi>n</mi><mo>+</mo><mn>2</mn></mrow></msub><mo>=</mo><msub><mi>F</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>+</mo><msub><mi>F</mi><mi>n</mi></msub></mrow></semantics></math></inline-formula>, for <inline-formula><math display="inline"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>0</mn></mrow></semantics></math></inline-formula>, where <inline-formula><math display="inline"><semantics><mrow><msub><mi>F</mi><mn>0</mn></msub><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula> and <inline-formula><math display="inline"><semantics><mrow><msub><mi>F</mi><mn>1</mn></msub><mo>=</mo><mn>1</mn></mrow></semantics></math></inline-formula>. There are several interesting identities involving this sequence such as <inline-formula><math display="inline"><semantics><mrow><msubsup><mi>F</mi><mi>n</mi><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>F</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><mn>2</mn></msubsup><mo>=</mo><msub><mi>F</mi><mrow><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow></semantics></math></inline-formula>, for all <inline-formula><math display="inline"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>0</mn></mrow></semantics></math></inline-formula>. In 2012, Chaves, Marques and Togbé proved that if <inline-formula><math display="inline"><semantics><msub><mrow><mo>(</mo><msub><mi>G</mi><mi>m</mi></msub><mo>)</mo></mrow><mi>m</mi></msub></semantics></math></inline-formula> is a linear recurrence sequence (under weak assumptions) and <inline-formula><math display="inline"><semantics><mrow><msubsup><mi>G</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><mi>s</mi></msubsup><mo>+</mo><mo>⋯</mo><mo>+</mo><msubsup><mi>G</mi><mrow><mi>n</mi><mo>+</mo><mo>ℓ</mo></mrow><mi>s</mi></msubsup><mo>∈</mo><msub><mrow><mo>(</mo><msub><mi>G</mi><mi>m</mi></msub><mo>)</mo></mrow><mi>m</mi></msub></mrow></semantics></math></inline-formula>, for infinitely many positive integers <i>n</i>, then <i>s</i> is bounded by an effectively computable constant depending only on <i>ł</i> and the parameters of <inline-formula><math display="inline"><semantics><msub><mrow><mo>(</mo><msub><mi>G</mi><mi>m</mi></msub><mo>)</mo></mrow><mi>m</mi></msub></semantics></math></inline-formula>. In this paper, we shall prove that if <inline-formula><math display="inline"><semantics><mrow><mi>P</mi><mo>(</mo><msub><mi>x</mi><mn>1</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>x</mi><mo>ℓ</mo></msub><mo>)</mo></mrow></semantics></math></inline-formula> is an integer homogeneous <i>s</i>-degree polynomial (under weak hypotheses) and if <inline-formula><math display="inline"><semantics><mrow><mi>P</mi><mrow><mo>(</mo><msub><mi>G</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>G</mi><mrow><mi>n</mi><mo>+</mo><mo>ℓ</mo></mrow></msub><mo>)</mo></mrow><mo>∈</mo><msub><mrow><mo>(</mo><msub><mi>G</mi><mi>m</mi></msub><mo>)</mo></mrow><mi>m</mi></msub></mrow></semantics></math></inline-formula> for infinitely many positive integers <i>n</i>, then <i>s</i> is bounded by an effectively computable constant depending only on <i>ℓ</i>, the parameters of <inline-formula><math display="inline"><semantics><msub><mrow><mo>(</mo><msub><mi>G</mi><mi>m</mi></msub><mo>)</mo></mrow><mi>m</mi></msub></semantics></math></inline-formula> and the coefficients of <i>P</i>.https://www.mdpi.com/2227-7390/8/12/2152homogeneous polynomiallinear forms in logarithmslinear recurrence sequence |
| spellingShingle | Marie Hubálovská Štěpán Hubálovský Eva Trojovská On Homogeneous Combinations of Linear Recurrence Sequences Mathematics homogeneous polynomial linear forms in logarithms linear recurrence sequence |
| title | On Homogeneous Combinations of Linear Recurrence Sequences |
| title_full | On Homogeneous Combinations of Linear Recurrence Sequences |
| title_fullStr | On Homogeneous Combinations of Linear Recurrence Sequences |
| title_full_unstemmed | On Homogeneous Combinations of Linear Recurrence Sequences |
| title_short | On Homogeneous Combinations of Linear Recurrence Sequences |
| title_sort | on homogeneous combinations of linear recurrence sequences |
| topic | homogeneous polynomial linear forms in logarithms linear recurrence sequence |
| url | https://www.mdpi.com/2227-7390/8/12/2152 |
| work_keys_str_mv | AT mariehubalovska onhomogeneouscombinationsoflinearrecurrencesequences AT stepanhubalovsky onhomogeneouscombinationsoflinearrecurrencesequences AT evatrojovska onhomogeneouscombinationsoflinearrecurrencesequences |