<p>In recent years, Sun has proposed numerous conjectures regarding the log-concavity of root sequences <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\{\root n \of {a_n}\}_{n\ge 1}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mo stretchy="false">{</mo> <mroot> <msub> <mi>a</mi> <mi>n</mi> </msub> <mi>n</mi> </mroot> <mo stretchy="false">}</mo> </mrow> <mrow> <mi>n</mi> <mo>≥</mo> <mn>1</mn> </mrow> </msub> </math></EquationSource> </InlineEquation>. We establish criteria for the asymptotic log-concavity of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\{\root n \of {a_n}\}_{n\ge 1}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mo stretchy="false">{</mo> <mroot> <msub> <mi>a</mi> <mi>n</mi> </msub> <mi>n</mi> </mroot> <mo stretchy="false">}</mo> </mrow> <mrow> <mi>n</mi> <mo>≥</mo> <mn>1</mn> </mrow> </msub> </math></EquationSource> </InlineEquation> and the asymptotic ratio log-convexity of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\{\root n \of {a_n}\}_{n\ge 1}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mo stretchy="false">{</mo> <mroot> <msub> <mi>a</mi> <mi>n</mi> </msub> <mi>n</mi> </mroot> <mo stretchy="false">}</mo> </mrow> <mrow> <mi>n</mi> <mo>≥</mo> <mn>1</mn> </mrow> </msub> </math></EquationSource> </InlineEquation> for P-recursive sequences <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\{a_n\}_{n\ge {1}}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mo stretchy="false">{</mo> <msub> <mi>a</mi> <mi>n</mi> </msub> <mo stretchy="false">}</mo> </mrow> <mrow> <mi>n</mi> <mo>≥</mo> <mn>1</mn> </mrow> </msub> </math></EquationSource> </InlineEquation>. Additionally, by the aid of symbolic computation, we present a systematic approach to determine the explicit integer <i>N</i> such that the sequence <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\{\root n \of {a_n}\}_{n\ge {N}}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mo stretchy="false">{</mo> <mroot> <msub> <mi>a</mi> <mi>n</mi> </msub> <mi>n</mi> </mroot> <mo stretchy="false">}</mo> </mrow> <mrow> <mi>n</mi> <mo>≥</mo> <mi>N</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> is log-concave and the sequence <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\{\root n \of {a_n}\}_{n\ge N}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mo stretchy="false">{</mo> <mroot> <msub> <mi>a</mi> <mi>n</mi> </msub> <mi>n</mi> </mroot> <mo stretchy="false">}</mo> </mrow> <mrow> <mi>n</mi> <mo>≥</mo> <mi>N</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> is ratio log-convex.</p>

Log-behavior of the root sequences of P-recursive sequences

期刊论文
发表日期：

2026年5月4日

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Qing-Hu Hou,
Zhongjie Li

摘要

In recent years, Sun has proposed numerous conjectures regarding the log-concavity of root sequences $\{\root n \of {a_n}\}_{n\ge 1}$ ${\sqrt[n]{a_{n}}}_{n \geq 1}$ . We establish criteria for the asymptotic log-concavity of $\{\root n \of {a_n}\}_{n\ge 1}$ ${\sqrt[n]{a_{n}}}_{n \geq 1}$ and the asymptotic ratio log-convexity of $\{\root n \of {a_n}\}_{n\ge 1}$ ${\sqrt[n]{a_{n}}}_{n \geq 1}$ for P-recursive sequences $\{a_n\}_{n\ge {1}}$ ${a_{n}}_{n \geq 1}$ . Additionally, by the aid of symbolic computation, we present a systematic approach to determine the explicit integer N such that the sequence $\{\root n \of {a_n}\}_{n\ge {N}}$ ${\sqrt[n]{a_{n}}}_{n \geq N}$ is log-concave and the sequence $\{\root n \of {a_n}\}_{n\ge N}$ ${\sqrt[n]{a_{n}}}_{n \geq N}$ is ratio log-convex.