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Abstract <p>In this article, let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(g(z)\)</EquationSource> <!--ContMath2560051Liu-m1--> </InlineEquation> be a meromorphic function of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\rho_{2}(g)&lt;1\)</EquationSource> <!--ContMath2560051Liu-m2--> </InlineEquation>. Suppose that a linear differential-difference polynomial of the form</p> <p><Equation ID="Equa"> <EquationSource Format="TEX">\(P(z,g)=\sum\limits_{k=1}^{n}{{b_{k}(z)}g(z+c_{k})}+\sum\limits_{k=1}^{m}{{d_{k}(z)}}{g^{(k)}}(z+\alpha_{k})+\sum\limits_{k=1}^{p}{{t_{k}(z)}}{g^{(k)}}(z),\)</EquationSource> <!--ContMath2560051Liu-m3--> </Equation></p> <p>where <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(n\)</EquationSource> <!--ContMath2560051Liu-m4--> </InlineEquation>, <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(m\)</EquationSource> <!--ContMath2560051Liu-m5--> </InlineEquation>, <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(p\geq 1\)</EquationSource> <!--ContMath2560051Liu-m6--> </InlineEquation> and <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(b_{k}(z)\)</EquationSource> <!--ContMath2560051Liu-m7--> </InlineEquation>, <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(d_{k}(z)\)</EquationSource> <!--ContMath2560051Liu-m8--> </InlineEquation>, <InlineEquation ID="IEq8"> <EquationSource Format="TEX">\(t_{k}(z)\)</EquationSource> <!--ContMath2560051Liu-m9--> </InlineEquation> are nonzero small functions relative to <InlineEquation ID="IEq9"> <EquationSource Format="TEX">\(g(z)\)</EquationSource> <!--ContMath2560051Liu-m10--> </InlineEquation>, and <InlineEquation ID="IEq10"> <EquationSource Format="TEX">\(c_{k}\)</EquationSource> <!--ContMath2560051Liu-m11--> </InlineEquation>, <InlineEquation ID="IEq11"> <EquationSource Format="TEX">\(\alpha_{k}\)</EquationSource> <!--ContMath2560051Liu-m12--> </InlineEquation> are distinct complex numbers. We study the uniqueness problems of <InlineEquation ID="IEq12"> <EquationSource Format="TEX">\(g(z)\)</EquationSource> <!--ContMath2560051Liu-m13--> </InlineEquation> and <InlineEquation ID="IEq13"> <EquationSource Format="TEX">\(P(z,g)\)</EquationSource> <!--ContMath2560051Liu-m14--> </InlineEquation>. Meantime we obtain some results related to the complex differential-difference equations with a more general form than the previous equations given by Zhang et al. [<CitationRef CitationID="CR1">1</CitationRef>].</p>

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Uniqueness on Linear Differential-Difference Polynomials

  • 期刊论文

  • 发表日期：

    2026年4月18日

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  • Y. Liu,
  • S. Zhong,
  • H. Wang

摘要

Abstract

In this article, let \(g(z)\) be a meromorphic function of \(\rho_{2}(g)<1\) . Suppose that a linear differential-difference polynomial of the form

\(P(z,g)=\sum\limits_{k=1}^{n}{{b_{k}(z)}g(z+c_{k})}+\sum\limits_{k=1}^{m}{{d_{k}(z)}}{g^{(k)}}(z+\alpha_{k})+\sum\limits_{k=1}^{p}{{t_{k}(z)}}{g^{(k)}}(z),\)

where \(n\) , \(m\) , \(p\geq 1\) and \(b_{k}(z)\) , \(d_{k}(z)\) , \(t_{k}(z)\) are nonzero small functions relative to \(g(z)\) , and \(c_{k}\) , \(\alpha_{k}\) are distinct complex numbers. We study the uniqueness problems of \(g(z)\) and \(P(z,g)\) . Meantime we obtain some results related to the complex differential-difference equations with a more general form than the previous equations given by Zhang et al. [1].