整函數(shù)及其微分多項(xiàng)式分擔(dān)一個(gè)多項(xiàng)式

張國(guó)威,陳昂

(1.安陽師范學(xué)院數(shù)學(xué)與統(tǒng)計(jì)學(xué)院,河南 安陽 455000;2.教育部考試中心,北京 100084)

整函數(shù)及其微分多項(xiàng)式分擔(dān)一個(gè)多項(xiàng)式

張國(guó)威1,陳昂2

(1.安陽師范學(xué)院數(shù)學(xué)與統(tǒng)計(jì)學(xué)院,河南 安陽 455000;2.教育部考試中心,北京 100084)

將Br¨uck猜想目前得到的幾個(gè)結(jié)論進(jìn)行了推廣,研究了整函數(shù)及其微分多項(xiàng)式分擔(dān)的一個(gè)多項(xiàng)式時(shí)的問題,并且得到了一個(gè)與之相關(guān)的復(fù)微分方程的解的性質(zhì).另外,還得到了一個(gè)定理,這個(gè)定理改進(jìn)了一些已知的結(jié)果.

整函數(shù);Nevanlinna理論;唯一性;分擔(dān)值

1 引言及主要結(jié)果

本文中使用了Nevanlinna經(jīng)典理論的一些基本符號(hào)和基本定理,關(guān)于這部分詳細(xì)內(nèi)容可見文獻(xiàn)[1-4].令f(z)和g(z)為復(fù)平面?上的兩個(gè)非常數(shù)亞純函數(shù),并且令P(z)為一個(gè)多項(xiàng)式或者是一個(gè)有限數(shù).用deg P(z)來定義多項(xiàng)式P(z)的級(jí).用f(z)=P(z)?g(z)=P(z)來表示:當(dāng)f(z)-P(z)=0時(shí)可推得g(z)-P(z)=0.若有f(z)=P(z)?g(z)=P(z)且g(z)=P(z)?f(z)=P(z),則將其表示為f(z)=P(z)?g(z)=P(z),并且稱f(z)和g(z)分擔(dān)P(z)IM(不計(jì)零點(diǎn)重?cái)?shù)).如果f(z)-P(z)和g(z)-P(z)有相同的零點(diǎn)并且這些零點(diǎn)的重?cái)?shù)相同,則稱f(z)和g(z)分擔(dān)P(z)CM(計(jì)零點(diǎn)重?cái)?shù))[1].更進(jìn)一步,用記號(hào)σ(f), ν(f)來定義f(z)的級(jí)和超級(jí).下面給出定義:

2 幾個(gè)引理

3 定理1.3的證明

證明將分兩種情況討論.

情況1如果P(z)是個(gè)多項(xiàng)式.如果f不是個(gè)超越的整函數(shù),由于方程(1)的解均為多項(xiàng)式,因此由方程(1),可知eP=c是個(gè)常數(shù),則ν(f)=σ(eP)=0,易知定理1.3的結(jié)論成立.

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Entire functions that share one polynomial with their linear differential polynomials

Zhang Guowei1,Chen Ang2

(1. School of Mathematics and Statistics, Anyang Normal University, Anyang 455000, China; 2. National Education Examinations Authority, Beijing 100084, China)

In this paper,we improve some known results about Bruck's conjecture. We study the problem that entire function and its linear differential polynomial share a polynomial and obtain some properties of solution of the related complex differential equation. Moreover, we get a theorem which improves some known results.

entire functions,Nevan linna theory,uniqueness,share value

O174.5

A

1008-5513(2012)02-0196-05

2010-12-10.

河南省教育廳重點(diǎn)項(xiàng)目(12A 110002).

張國(guó)威(1981-),博士,講師,研究方向：值分布論,復(fù)微分方程,復(fù)動(dòng)力系統(tǒng)等.

2010 MSC:30D 35