<p>In this paper we investigate analytic properties of forcing and anti-forcing polynomials of pyrene chains. We show that zeros of the forcing polynomials <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$F(P_n, x)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>F</mi> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mi>n</mi> </msub> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> are real, located in the interval <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$(-1, 0]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mo>-</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation>, and dense in the corresponding closed interval. We also show that the coefficients of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$F(P_n, x)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>F</mi> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mi>n</mi> </msub> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> are unimodal, log-concave, and asymptotically normal. For anti-forcing polynomials <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$Af(P_n, x)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mi>n</mi> </msub> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, we show that the coefficients are unimodal and log-concave. Furthermore, the distribution of zeros of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$Af(P_n, x)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mi>n</mi> </msub> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is also investigated.</p>

Analytic Properties of Forcing and Anti-Forcing Polynomials of Pyrene Chains

期刊论文
发表日期：

2026年2月14日

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Guanru Li,
Guanwu Liu

摘要

In this paper we investigate analytic properties of forcing and anti-forcing polynomials of pyrene chains. We show that zeros of the forcing polynomials $F(P_n, x)$ $F (P_{n}, x)$ are real, located in the interval $(-1, 0]$ $(- 1, 0]$ , and dense in the corresponding closed interval. We also show that the coefficients of $F(P_n, x)$ $F (P_{n}, x)$ are unimodal, log-concave, and asymptotically normal. For anti-forcing polynomials $Af(P_n, x)$ $A f (P_{n}, x)$ , we show that the coefficients are unimodal and log-concave. Furthermore, the distribution of zeros of $Af(P_n, x)$ $A f (P_{n}, x)$ is also investigated.