On Log-Definite Tempered Combinatorial Sequences
This article is concerned with qualitative and quantitative refinements of the concepts of the log-convexity and log-concavity of positive sequences. A new class of tempered sequences is introduced, its basic properties are established and several interesting examples are provided. The new class ext...
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MDPI AG
2025-04-01
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| author | Tomislav Došlić Biserka Kolarec |
| author_facet | Tomislav Došlić Biserka Kolarec |
| author_sort | Tomislav Došlić |
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| description | This article is concerned with qualitative and quantitative refinements of the concepts of the log-convexity and log-concavity of positive sequences. A new class of tempered sequences is introduced, its basic properties are established and several interesting examples are provided. The new class extends the class of log-balanced sequences by including the sequences of similar growth rates, but of the opposite log-behavior. Special attention is paid to the sequences defined by two- and three-term linear recurrences with constant coefficients. For the special cases of generalized Fibonacci and Lucas sequences, we graphically illustrate the domains of their log-convexity and log-concavity. For an application, we establish the concyclicity of the points <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mfenced separators="" open="(" close=")"><mstyle scriptlevel="0" displaystyle="true"><mfrac><msub><mi>a</mi><mrow><mn>2</mn><mi>n</mi></mrow></msub><msub><mi>a</mi><mrow><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mfrac></mstyle><mo>,</mo><mstyle scriptlevel="0" displaystyle="true"><mfrac><mn>1</mn><msub><mi>a</mi><mrow><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mfrac></mstyle></mfenced></semantics></math></inline-formula> for some classes of Horadam sequences <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>a</mi><mi>n</mi></msub><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula> with positive terms. |
| format | Article |
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| institution | DOAJ |
| issn | 2227-7390 |
| language | English |
| publishDate | 2025-04-01 |
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| series | Mathematics |
| spelling | doaj-art-cf12afbaa48c43dbb750d4cf6afe3f022025-08-20T03:06:27ZengMDPI AGMathematics2227-73902025-04-01137117910.3390/math13071179On Log-Definite Tempered Combinatorial SequencesTomislav Došlić0Biserka Kolarec1Department of Mathematics, Faculty of Civil Engineering, University of Zagreb, Kačićeva ulica 26, 10 000 Zagreb, CroatiaDepartment of Information Science and Mathematics, Faculty of Agriculture, University of Zagreb, Svetošimunska cesta 25, 10 000 Zagreb, CroatiaThis article is concerned with qualitative and quantitative refinements of the concepts of the log-convexity and log-concavity of positive sequences. A new class of tempered sequences is introduced, its basic properties are established and several interesting examples are provided. The new class extends the class of log-balanced sequences by including the sequences of similar growth rates, but of the opposite log-behavior. Special attention is paid to the sequences defined by two- and three-term linear recurrences with constant coefficients. For the special cases of generalized Fibonacci and Lucas sequences, we graphically illustrate the domains of their log-convexity and log-concavity. For an application, we establish the concyclicity of the points <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mfenced separators="" open="(" close=")"><mstyle scriptlevel="0" displaystyle="true"><mfrac><msub><mi>a</mi><mrow><mn>2</mn><mi>n</mi></mrow></msub><msub><mi>a</mi><mrow><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mfrac></mstyle><mo>,</mo><mstyle scriptlevel="0" displaystyle="true"><mfrac><mn>1</mn><msub><mi>a</mi><mrow><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mfrac></mstyle></mfenced></semantics></math></inline-formula> for some classes of Horadam sequences <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>a</mi><mi>n</mi></msub><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula> with positive terms.https://www.mdpi.com/2227-7390/13/7/1179tempered sequencelog-convexitylog-concavitylog-balancednessFibonacci numbersLucas numbers |
| spellingShingle | Tomislav Došlić Biserka Kolarec On Log-Definite Tempered Combinatorial Sequences Mathematics tempered sequence log-convexity log-concavity log-balancedness Fibonacci numbers Lucas numbers |
| title | On Log-Definite Tempered Combinatorial Sequences |
| title_full | On Log-Definite Tempered Combinatorial Sequences |
| title_fullStr | On Log-Definite Tempered Combinatorial Sequences |
| title_full_unstemmed | On Log-Definite Tempered Combinatorial Sequences |
| title_short | On Log-Definite Tempered Combinatorial Sequences |
| title_sort | on log definite tempered combinatorial sequences |
| topic | tempered sequence log-convexity log-concavity log-balancedness Fibonacci numbers Lucas numbers |
| url | https://www.mdpi.com/2227-7390/13/7/1179 |
| work_keys_str_mv | AT tomislavdoslic onlogdefinitetemperedcombinatorialsequences AT biserkakolarec onlogdefinitetemperedcombinatorialsequences |