This paper studies value distribution of entire function f(z) and their forward differences <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="9_2025_2838_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="53" /> </InlineMediaObject> <EquationSource Format="TEX">$\Delta ^{n}f(z)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi mathvariant="normal">Δ</mi> <mi>n</mi> </msup> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. We mainly prove: let <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="9_2025_2838_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="45" /> </InlineMediaObject> <EquationSource Format="TEX">$n\in {\mathbb {N}}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>∈</mo> <mi mathvariant="double-struck">N</mi> </mrow> </math></EquationSource> </InlineEquation>, f(z) be a transcendental entire function such that f(z) and <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="9_2025_2838_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="53" /> </InlineMediaObject> <EquationSource Format="TEX">$\Delta ^{n}f(z)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi mathvariant="normal">Δ</mi> <mi>n</mi> </msup> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> have finitely many zeros without growth condition, then the exact forms of f(z) and <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="9_2025_2838_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="53" /> </InlineMediaObject> <EquationSource Format="TEX">$\Delta ^{n}f(z)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi mathvariant="normal">Δ</mi> <mi>n</mi> </msup> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> are given.

Picard Values of Entire Functions and Their Differences

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2025年4月15日

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Shuang-Ting Lan,
Zong-Xuan Chen

摘要

This paper studies value distribution of entire function f(z) and their forward differences $\Delta ^{n}f(z)$ $Δ^{n} f (z)$ . We mainly prove: let $n\in {\mathbb {N}}$ $n \in N$ , f(z) be a transcendental entire function such that f(z) and $\Delta ^{n}f(z)$ $Δ^{n} f (z)$ have finitely many zeros without growth condition, then the exact forms of f(z) and $\Delta ^{n}f(z)$ $Δ^{n} f (z)$ are given.