On Log-Definite Tempered Combinatorial Sequences - Tomislav Došlić; Biserka Kolarec (Mathematics)
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 <semantics><mfenced separators="" open="(" close=")"><mstyle scriptlevel="0" displaystyle="true">a2na2n+1</mstyle>,<mstyle scriptlevel="0" displaystyle="true">1a2n+1</mstyle></mfenced></semantics> for some classes of Horadam sequences <semantics>(an)</semantics> with positive terms.