Recently, some unicity results about the <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41478_2025_902_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="24" /> </InlineMediaObject> <EquationSource Format="TEX">$k^{th}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>k</mi> <mrow> <mi mathvariant="italic">th</mi> </mrow> </msup> </math></EquationSource> </InlineEquation> derivative and shift <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41478_2025_902_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="61" /> </InlineMediaObject> <EquationSource Format="TEX">$f(z+c)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo>+</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> of a meromorphic function f(z) have developed. In this paper, we continue to study this topic. We mainly prove that for a meromorphic function of hyper-order strictly less than 1, if <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41478_2025_902_Article_IEq3.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">$L_{f}(z)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>L</mi> <mi>f</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, the linear differential polynomial of f and <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41478_2025_902_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="61" /> </InlineMediaObject> <EquationSource Format="TEX">$f(z+c)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo>+</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> share <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41478_2025_902_Article_IEq5.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">$\infty$</EquationSource> <EquationSource Format="MATHML"><math> <mi>∞</mi> </math></EquationSource> </InlineEquation> CM along with three partially shared values, then <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41478_2025_902_Article_IEq6.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="123" /> </InlineMediaObject> <EquationSource Format="TEX">$f(z+c)= L_{f}(z)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo>+</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msub> <mi>L</mi> <mi>f</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. Outcomes of this paper generalize the results due to Kaish and Mondal (J Anal 31(1):329–342, 2023).

Linear differential polynomial of a meromorphic function sharing values with shift

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发表日期：

2025年4月23日

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Dilip Chandra Pramanik,
Nimai Dey

摘要

Recently, some unicity results about the $k^{th}$ $k^{th}$ derivative and shift $f(z+c)$ $f (z + c)$ of a meromorphic function f(z) have developed. In this paper, we continue to study this topic. We mainly prove that for a meromorphic function of hyper-order strictly less than 1, if $L_{f}(z)$ $L_{f} (z)$ , the linear differential polynomial of f and $f(z+c)$ $f (z + c)$ share $\infty$ $\infty$ CM along with three partially shared values, then $f(z+c)= L_{f}(z)$ $f (z + c) = L_{f} (z)$ . Outcomes of this paper generalize the results due to Kaish and Mondal (J Anal 31(1):329–342, 2023).