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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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2025-04-01
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| Series: | Mathematics |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2227-7390/13/7/1179 |
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| Summary: | 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. |
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| ISSN: | 2227-7390 |