陳宗煊, 黃志波

(華南師范大學(xué)數(shù)學(xué)科學(xué)學(xué)院,廣東廣州 510631)

復(fù)域差分和差分方程的研究

陳宗煊*, 黃志波

(華南師范大學(xué)數(shù)學(xué)科學(xué)學(xué)院,廣東廣州 510631)

介紹了近十年來復(fù)域差分、q-差分、差分方程及q-差分方程的主要研究成果,其中包括亞純函數(shù)對數(shù)導(dǎo)數(shù)引理的差分模擬;Clunie引理和Mohon′ko引理的差分模擬; 慢增長亞純函數(shù)的差分、均差分的零點和不動點的性質(zhì); 差分多項式的值分布性質(zhì);差分Riccati方程與差分Painlevé方程亞純解的性質(zhì);復(fù)域q-差分及q-差分方程的解析性質(zhì).

復(fù)域差分; 差分方程; 復(fù)域q-差分;q-差分方程; 亞純函數(shù)值分布

1925年,NEVANLINNA[1]發(fā)表了關(guān)于亞純函數(shù)理論的論文,后來發(fā)展為亞純函數(shù)Nevanlinna理論.隨后, 亞純函數(shù)Nevanlinna理論被運用到線性和非線性微分方程亞純解的值分布、唯一性和存在性等問題的討論, 獲得豐富的成果[2-8].

然而,利用亞純函數(shù)Nevanlinna理論,對差分方程的研究可以追溯到二十世紀(jì)初期[9-10].由于缺乏有力的研究工具,復(fù)域差分方程理論的發(fā)展極其緩慢.雖然在20世紀(jì)七、八十年代,BANK和KAUFMAN[11], SHIMOMURA[12]和YANAGIHARA[13]獲得一些關(guān)于差分方程亞純解存在性的初始結(jié)果. 特別地,BANK和KAUFMAN[11]證明了, 對任意關(guān)于z的有理函數(shù)R(z),差分方程

f(z+1)-f(z)=R(z)

(1)

總存在滿足條件T(r,f)=O(r)的亞純解f(z). YANAGIHARA[13]證明了, 對任意關(guān)于f(z)的有理函數(shù)R(f(z)),差分方程

f(z+1)=R(f(z))

(2)

有非平凡的亞純解.

2000年, ABLOWITZ等[14]利用亞純函數(shù)Nevanlinna 理論研究二階非線性差分方程的可積性問題, 標(biāo)志著亞純函數(shù)Nevanlinna 理論作用到復(fù)域差分方程研究的真正實現(xiàn). 隨后,經(jīng)過許多復(fù)分析和數(shù)學(xué)物理領(lǐng)域的專家和學(xué)者近十幾年的努力, 亞純函數(shù)Nevanlinna 理論的差分模擬的研究取得重大突破,從而為復(fù)域差分方程的研究提供了有力的理論工具. 由此,復(fù)域差分、q-差分、復(fù)域差分方程及q-差分方程的研究,以及與之對應(yīng)的唯一性理論的差分模擬的研究,逐漸成為熱門研究課題.

近十年來,亞純函數(shù)Nevanlinna理論為復(fù)域差分、q-差分、復(fù)域差分方程及q-差分方程的研究和發(fā)展提供了一個有力的工具,主要體現(xiàn)在亞純函數(shù)的對數(shù)導(dǎo)數(shù)引理的差分模擬和Clunie-Mohon′ko引理的差分模擬. 下面將介紹其差分形式和q-差分形式.

有限級亞純函數(shù)的對數(shù)導(dǎo)數(shù)引理的差分模擬, 是由CHIANG和FENG[15]、HALBURD和KORHONEN[16-17]分別建立的. 這里, 我們給出它的最終形式.

對所有的r成立,至多除去一個對數(shù)測度為有限的集合.

對于有限級亞純函數(shù)f(z)的差分算子, HALBURD和KORHONEN[17]得到如下的對數(shù)導(dǎo)數(shù)引理.

對所有的r成立, 至多除去一個對數(shù)測度為有限的集合.

如果f(z)為無窮級的亞純函數(shù), HALBURD等[18]和KORHONEN[19]獲得對數(shù)導(dǎo)數(shù)引理的差分模擬的更一般的形式.

對所有的r成立,至多除去一個集合E滿足

如果f(z)的超級σ2(f)<1和ε>0,那么

對所有的r成立, 至多除去一個對數(shù)測度為有限的集合.

定理4[19]設(shè)f(z)為非常數(shù)亞純函數(shù),ω(z)=czn+pn-1zn-1+…+p0和φ(z)=czn+qn-1zn-1+…+q0是2個非常數(shù)多項式. 如果

那么

對所有的r成立,至多除去一個對數(shù)測度為有限的集合.

Clunie引理和Mohon′ko引理在微分多項式的值分布和微分方程亞純解的增長級估計方面具有重要的作用.Clunie 引理的差分模擬最初形式可以參考文獻[16]的定理3.1和文獻[20]的定理2.3,對于更進一步的結(jié)果, 可以參考文獻[21]-[23]. 下面給出其最初結(jié)論.

定理5 設(shè)f(z)是增長級為σ(<∞)的亞純函數(shù),且滿足方程

U(z,f)P(z,f)=Q(z,f),

其中U(z,f),P(z,f)和Q(z,f)是差分多項式滿足degfQ(z,f)≤degfU(z,f)=n. 更進一步地,U(z,f)僅含一項次數(shù)最大的項. 那么,對任意ε>0,

m(P(z,f))=O(rσ-1+ε)+S(r,f)

對所有的r成立, 至多除去一個對數(shù)測度為有限的集合.

下面給出Mohon′ko引理的差分模擬.

定理6[16,20]設(shè)f(z)是有限級亞純函數(shù),且滿足方程P(z,f)=0,其中P(z,f)是差分多項式.對給定的小函數(shù)a(z),如果P(z,a(z))?0, 那么

下面將給出對數(shù)導(dǎo)數(shù)引理、Clunie引理和Mohon′ko引理的q-差分模擬.

在一對數(shù)密度為1的集合上成立.

定理8[24]設(shè)f(z)是非常數(shù)零級亞純函數(shù)且滿足fn(z)P(z,f)=Q(z,f),其中P(z,f)和Q(z,f)是關(guān)于f(z)的q-差分多項式且degfQ(z,f)≤n,那么

m(r,P(z,f))=o(T(r,f))

在一對數(shù)密度為1的集合上成立.

定理9[24]設(shè)f(z)是非常數(shù)零級亞純函數(shù)且滿足P(z,f)=0,其中P(z,f)是關(guān)于f(z)的q-差分多項式.對給定的小函數(shù)a(z),如果P(z,a(z))?0, 那么

在一對數(shù)密度為1的集合上成立.

對于亞純函數(shù)f(z),其位移f(z+c)和q-位移f(qz)的特征函數(shù)等的估計[15,25]、Nevanlinna第二基本定理[17,24]、Wiman-Valiron方法的差分模擬[26]等,這里不再闡述. 基于上述理論的建立,復(fù)域差分、q-差分、差分方程及q-差分方程和唯一性理論的差分模擬取得豐富的研究成果.

1 復(fù)域差分多項式

鑒于對數(shù)導(dǎo)數(shù)引理和Clunie-Mohon′ko引理等差分模擬的建立,對復(fù)域差分多項式的解析性質(zhì)的研究,主要體現(xiàn)在差分多項式的值分布和唯一性兩方面.

1.1復(fù)域差分多項式的值分布

具有無窮多個零點.

同時提出了如下猜想：

猜想1[27]假設(shè)f(z)是超越整函數(shù)且滿足σ(f)<1,那么G(z)有無窮多個零點.

若所考慮的函數(shù)f(z)為亞純函數(shù),BERGWEILER和LANGLEY還證明了:

隨后，LANGLEY[28]、CHEN和SHON[29]推廣了上述結(jié)果,并給出很多新穎的證明方法.特別地,CHEN和SHON[29]首次研究了具有慢增長性級的亞純函數(shù)的差分和均差分的不動點問題, 從而部分地證明了上述猜想.

具有無窮多個零點和無窮多個不動點.

(i)至多有有限個極點zj,zk滿足zj-zk=c;

那么

具有無窮多個零點和無窮多個不動點.

對于差分多項式的值分布的研究, 要歸結(jié)于對HAYMAN[30]所探究的微分多項式值分布的差分模擬[31-33]. 隨后, 亞純函數(shù)的差分多項式的值分布獲得一系列有趣的結(jié)果[23,33-42]. 下面介紹HAYMAN關(guān)于微分多項式的幾個經(jīng)典結(jié)果的差分模擬.

例1[32]設(shè)f(z)=ez+1,則

H(z)=f(z)f(z+iπ)-1=(1+ez)(1-ez)-1=-e2z.

這個例子說明,定理14和定理15中的條件n≥2不能省略. 因此,CHEN等進一步獲得如下結(jié)果.

1.2復(fù)域差分多項式的唯一性

設(shè)f(z)和g(z)為2個非常數(shù)的亞純函數(shù),a為任意復(fù)數(shù). 如果f(z)-a和g(z)-a具有相同的零點(計算重數(shù)), 則稱f(z)和g(z)具有CM分擔(dān)a. 如果f(z)-a和g(z)-a具有相同的零點(不計算重數(shù)),則稱f(z)和g(z)具有IM分擔(dān)a. 亞純函數(shù)唯一性理論的典型結(jié)果是Nevanlinna五值定理和四值定理[8]. 亞純函數(shù)唯一性理論的已有文獻指出,對于任意2個亞純函數(shù), 如果其分擔(dān)值的個數(shù)減少, 需要增加額外的假設(shè).基于這一事實,HEITTOKANGAS等[43-44]首先研究了亞純函數(shù)f(z)和它的位移算子f(z+c)的唯一性,后來進一步提升已有的結(jié)果, 得到

近幾年來, 很多學(xué)者在這一方面做出很多有意義的結(jié)果[42,45-54].

關(guān)于亞純函數(shù)唯一性理論的差分模擬, Brück猜想[55]的差分模擬具有重要的地位. 下面介紹幾個經(jīng)典的結(jié)果,更多的結(jié)果可參考文獻[33]、[44]、[48]-[49]、[56].

其中A為一非零常數(shù).

2 復(fù)域差分方程

復(fù)域微分方程的復(fù)振蕩理論在過去30多年發(fā)展迅速, 獲得一系列的成果[2,5-6].這些成果揭示了微分方程亞純解的增長級和其系數(shù)增長級之間的關(guān)系,刻畫了微分方程亞純解的零點、 極點和不動點等性質(zhì). 近十年來,隨著亞純函數(shù)Nevanlinna理論的差分模擬的建立,對函數(shù)差分方程亞純解性質(zhì)的研究, 也進入一個繁榮時期.

2.1復(fù)域差分方程

ABLOWITZ等[14]考慮復(fù)域離散方程作為復(fù)域時滯方程,運用亞純函數(shù)Nevanlinna理論對其研究,為復(fù)域差分方程的研究提供了有力的工具.隨后,HALBURD和KORHONEN[57]利用奇異測試的方法證明了:如果二階差分方程

f(z+1)+f(z-1)=R(z,f)

(3)

存在有限級亞純解f(z),那么f(z)或者滿足差分Riccati方程

或者可經(jīng)過一線性變換,將方程(3)轉(zhuǎn)化為一些典型的線性差分方程和差分Painlevé方程.

2.1.1 復(fù)域差分方程亞純解的存在性 對復(fù)域差分方程亞純解存在性的研究,主要體現(xiàn)在有理解的存在性及其表示[58-61]、漸近解的存在性[57，61-62]和亞純解的表示[15,63-66]. 下面列出一些典型結(jié)果.

定理21[58]設(shè)a,b,c是常數(shù)且a,b不全為零.

(i)如果a≠0,那么差分Painlevé I方程

(4)

沒有有理解;

(ii)如果a=0和b≠0,那么差分Painlevé I方程(4)有一個非零常數(shù)解f(z)=A滿足

2A2-cA-b=0,

其他有理解具有形式f(z)=P(z)/Q(z),其中P(z)和Q(z)是互素的多項式且其次數(shù)滿足degP

定理22[59]設(shè)δ=±1,A(z)=m(z)/n(z)是不可約的有理函數(shù),其中m(z)和n(z)是多項式,其次數(shù)分別為degm(z)=m和degn(z)=n.

(i)假設(shè)m≥n且m-n是偶數(shù)或零.如果差分Riccati方程

(5)

有一個不可約的有理解f(z)=P(z)/Q(z),其中P(z)和Q(z)是多項式且其系數(shù)分別為degP(z)=p和degQ(z)=q, 那么p-q=(m-n)/2;

(ii)假設(shè)m≤n且m-n=k(≥2)是一正整數(shù).如果差分Riccati方程(5)有一個不可約的有理解f(z)=P(z)/Q(z),那么q-p=k-1或q-p=1;

(iii)假設(shè)m>n且m-n是奇數(shù),或m

SHIMOMURA[61]考慮差分Painlevé方程

(6)

(7)

(8)

(9)

定理23[61]如果α≠0,那么差分方程(6)～(8)沒有有理解．

對于一階非線性差分方程

f(z+1)=R(f(z)),

(10)

其中R(f(z))是關(guān)于f(z)的常系數(shù)有理函數(shù),HALBURD和KORHONEN[62]獲得方程(10)非平凡亞純解滿足一定的漸近性質(zhì).

(f(z)-γ)λ-z→α, Rez→-∞.

對于差分方程亞純解的表示形式,ISHIZAKI[63]獲得差分Riccati方程亞純解的表示性質(zhì).

定理26[63]設(shè)A(z)為一亞純函數(shù).如果差分Riccati方程

(11)

存在3個不同的亞純解f1(z)、f2(z)和f3(z), 那么方程(11)的任一亞純解f(z)可以表示為

f(z)=

(12)

其中Q(z)是周期為1的亞純函數(shù). 反之,對于任意周期為1的亞純函數(shù), 我們定義函數(shù)f(z)具有式(12)的形式,那么f(z)是方程(11)的亞純解.

2.1.2 復(fù)域差分方程亞純解的值分布 對于差分方程(包括:一般的差分方程、差分Painlevé方程和差分Riccati方程)亞純解的值分布,主要體現(xiàn)在差分方程亞純解的增長級、零點收斂指數(shù)、極點收斂指數(shù)、不動點收斂指數(shù)和Borel例外值的存在性, 詳見文獻[11]、[15]、[20]、[57]-[60]、[62]、[64]-[71]等, 這里不再贅述.

2.2復(fù)域函數(shù)方程

復(fù)域q-差分方程(函數(shù)方程)的研究可追溯到RITT[72]研究的自治Schr?der方程f(qz)=R(f(z)),其中R(f(z))是關(guān)于f(z)的常系數(shù)有理函數(shù),和VALIRON[73]研究的非自治Schr?der方程f(qz)=R(z,f(z)),其中R(z,f(z))是關(guān)于z、f(z)的有理函數(shù),它們是與復(fù)動力系統(tǒng)緊密聯(lián)系的方程[73-76].

2.2.1 復(fù)域q-差分方程亞純解的存在性 具有有理系數(shù)的線性q-差分方程(函數(shù)方程), 其亞純解不一定存在. 近十年來,BEIGWEILER、GUNDERSEN和HEITTOKANGAS等[77-79]在一定的系數(shù)約束下,研究了q-差分方程亞純解的存在性.下面給出一些經(jīng)典結(jié)果.

定理27[78]設(shè)f(z)是q-差分方程

f(qz)=A(z)+γf(z)+δf(z)2,

(13)

(i)在方程(13)中,如果

那么方程(13)有2個不同的亞純解;

(ii)在方程(13)中, 如果(1-γ)2=4,那么方程(13)只有一個亞純解.

定理28[79]設(shè)a0(z),a1(z),…,an(z)是復(fù)常數(shù)滿足

和Q(z)=g1(z)是一整函數(shù).那么q-差分方程

(14)

只有唯一的整函數(shù)解.

定理29[77]考慮q-差分方程

f(q2z)+a(z)f(qz)+b(z)f(z)=0,

(15)

(i)如果不存在整數(shù)n滿足q2n+a0qn+b0=0, 那么方程(15)沒有任何超越亞純解;

(ii)如果b0≠0且存在整數(shù)n滿足q2n+a0qn+b0=0,那么方程(15)存在一個超越亞純解;

(iii)如果b0=0,那么方程(15)沒有任何超越亞純解.

對于q-差分方程亞純解的表示形式,HUANG[80]獲得q-差分Riccati方程亞純解的表示性質(zhì):

(16)

存在3個不同的亞純解g1(z)、g2(z)和g3(z),那么方程(16)的任一亞純解f(z)可以表示為

g(z)=

(17)

其中φ(z)是亞純函數(shù)且滿足φ(qz)=φ(z).反之,對于任意亞純函數(shù)φ(z)且滿足φ(qz)=φ(z),我們定義函數(shù)f(z)具有式(17)的形式,那么f(z)是方程(16)的亞純解.

2.2.2 復(fù)域q-差分方程亞純解的值分布 隨著亞純函數(shù)q-差分模擬的建立,q-差分方程亞純解的值分布性質(zhì)也得到廣泛探討, 詳見文獻[47]、[77]-[86]等, 這里不再贅述.

3 結(jié)束語

亞純函數(shù)Nevanlinna理論差分模擬的建立,為復(fù)域(q-)差分多項式和復(fù)域(q-)差分方程的發(fā)展開辟了新天地.但由于對復(fù)域差分方程的研究,沒有普適的研究方法,針對不同形式的差分方程,我們需要尋求其特有的解法,特別是差分方程亞純解的存在性方面,還有大量的工作需要去探討.

[1] NEVANLINNA R. Zur theorie der meromorphen funktionen (German)[J].Acta math,1925,46(1-2):1-99.

[2] GAO S A,CHEN Z X,CHEN T W. Complex oscillation theory of linear differential equations[M].Wuhan: Huazhong Univ Sci Tech Press, 1998.

[3] GROMAK V,LAINE I,SHIMOMURA S.Painlevé differential equations in the complex plane[M].Berlin-New York:Walter de Gruyter,2002.

[4] HAYMAN W K. Meromorphic functions[M].Oxford:Clarendon Press,1964.

[5] HILLE E.Ordinary differential equations in the complex domain[M].Mineola, NY:Dover Publications, Inc,1997.

[6] LAINE I. Nevanlinna theory and complex differential equations[M].Berlin:Walter de Gruyter,1993.

[7] YANG L.Value distribution theorem and new research[M]. Beijing: Science Press, 1982.

[8] YANG C C,YI H X. Uniqueness theory of meromorphic functions[M]. Dordrecht: Kluwer Academic Publishers Group, 2003.

[9] BIRKHOFF G D.General theory of linear difference equations[J].Trans Amer Math Soc,1911,12(2):243-283.

[10] CARMICHAEL R D.Linear difference equations and their anaytic solutions[J].Trans Amer Math Soc,1911,12(1):99-134.

[11] BANK S B,KAUFMAN R P. An extension of H?lder′s theorem concerning the gamma function[J]. Funkcial Ekvac, 1976,19(1):53-63.

[12] SHIMOMURA S. Entire solutions of a polynomial difference equation[J].J Fac Sci Uni Tokyo Sect IA Math,1981,28(2):253-266.

[13] YANAGIHARA N. Meromorphic solutions of some difference equations[J].Funkcial Ekvac,1980,23(2):309-326.

[14] ABLOWITZ M J,HALBURD R,HERBST B. On the extension of the Painlevé property to difference equations[J]. Nonlinearity, 2000, 13(3):889-905.

[15] CHIANG Y M,FENG S J. On the Nevanlinna characteristic of and difference equations in the complex plane[J]. Ramanujan J,2008,16(1):105-129.

[16] HALBURD R G,KORHONEN R J. Difference analogue of the lemma on the logarithmic derivative with applications to difference equations[J].J Math Anal Appl,2006,314(2):477-487.

[17] HALBURD R G,KORHONEN R J. Nevanlinna theory for the difference operator[J]. Ann Acad Sci Fenn Math,2006,31(2):463-478.

[18] HALBURD R G,KORHONEN R J,TOGHN K.Holomorphic curves with shift invariant hyperplane preimages[J].arXiv:0903.3236.

[19] KORHONEN R.An extension of Picard′s theorem for meromorphic functions of small hyper order[J].J Math Anal Appl,2009,357(1):244-253.

[20] LAINE I,YANG C C. Clunie theorems for difference and q-difference equations[J].J London Math Soc,2007,76(3):556-566.

[21] HUANG Z B,CHEN Z X. A Clunie lemma for difference and q-difference polynomials[J].Bull Aust Math Soc,2010,81(1):23-32.

[22] KORHONEN R. A new Clunie type theorem for difference polynomials[J].J Difference Equ Appl,2011,17(3):387-400.

[23] ZHANG R R,CHEN Z X. Value distribution of difference polynomials of meromorphic functions[J]. Sci China:Ser A,2012, 42(11):1115-1130.

[24] BARNETT D C,HALBURD R G,MORGAN W,et al. Nevanlinna theory for the q-difference operator and meromorphic solutions of q-difference equations[J].Proc Roy Soc Edinburgh:Sec A,2007,137(3):457-474.

[25] ZHANG J L,KORHONEN R. On the Nevanlinna characteristic off(qz) and its applications[J].J Math Anal Appl,2010,369(2):537-544.

[26] ISHIZAKI K,YANAGIIHARA N.Wiman-Valiron method for difference equations[J].Nagoya Math J,2004,175:75-102.

[27] BERGWEILER W,LANGLEY J K.Zeros of meromorphic functions[J].Math Proc Camb Phil Soc,2007,142(1):133-147.

[28] LANGLEY J K. Value distribution of difference of meromorphic functions[J].Rocky Mountain J Math,2011,41(1):275-291.

[29] CHEN Z X,SHON K H. On zeros and fixed points of differences of meromorphic functions[J]. J Math Anal Appl, 2008, 344(1):373-383.

[30] HAYMAN W K.Picard values of meromorphic functions and their derivatives[J].Ann Math,1959,70(2):9-42.

[31] CHEN Z X,HUANG Z B,ZHENG X M. On properties of difference polynomials[J].Acta Math Sci:Ser B,2011,31(2):627-633.

[32] LAINE I,YANG C C.Value distribution of difference polynomials[J].Proc Japan Acad:Ser A,2007,83(8):148-151.

[33] LIU K,YANG L Z. Value distribution of their difference operator[J].Arch Math,2009,92(3):270-278.

[34] CHEN Z X. On value distribution of difference polynomials of meromorphic functions[J]. Abstr Appl Anal,2011,Art.ID 239853, 9pp.

[35] CHEN Z X,SHON K H.Properties of differences of meromorphic functions[J].Czechoslovak Math J,2011,61(1):213-224.

[36] CHEN Z X. Value distribution of products of meromorphic functions and their differences[J]. Taiwanese J Math,2011,15(4):1411-1421.

[37] CHEN Z X,SHON K H. Estimates for the zeros of difference of meromorphic functions[J]. Sci China:Ser A, 2009, 52(11):2447-2458.

[38] LIU K,LAINE I. A note on value distribution of difference polynomials[J]. Bull Aust Math Soc,2010,81(3):353-360.

[39] LIU Y,YI H X. On zeros of differences of meromorphic functions[J]. Ann Polon Math,2011,100(2):167-178.

[40] LIU Y,YI H X. On growth and zeros of difference of some meromorphic funtions[J].Ann Polon Math,2013,108(3):305-318.

[41] ZHANG R R,CHEN Z X.Value distribution of meromorphic functions and their differences[J].Turkish J Math,2012,36(2):395-406.

[42] ZHANG J L.Value distribution and shared set of differences of meromorphic functions[J].J Math Anal Appl,2010,367(2):401-408.

[43] HEITTOKANGAS J,KORHONEN R,LAINE I,et al.Uniqueness of meromorphic functions sharing values with their shifts[J]. Complex Var Elliptic Equ,2011,56(1-4):81-92.

[44] HEITTOKANGAS J,KORHONEN R,LAINE I,et al.Value sharing results for shifts of meromorphic functions, and sufficient condition for periodicity[J].J Math Anal Appl,2009,355(1):352-363.

[45] CHEN B Q,CHEN Z X.Meromorphic function sharing two sets with itts difference operator[J]. B Malays Math Sci So,2012,35(3):765-774.

[46] CHEN B Q,CHEN Z X,LI S.Uniqueness of difference operators of meromorphic functions[J]. J Inequal Appl,2012,48:1-9.

[47] HUANG Z B. Value distribution and uniqueness on q-differences of meromorphic functions[J].Bull Korean Math Soc,2013,50(4):1157-1171.

[48] LI S,GAO Z S. A note on the Brück conjecture[J]. Arch Math,2010,95(3):257-268.

[49] LI X M,KANG C Y,YI H X. Uniqueness theorems of entire functions sharing a nonzero complex number with their difference operators[J]. Arch Math,2011,96(6):577-587.

[50] LI X M,YI H X,KANG C Y. Notes on entire functions sharing an entire function of a small order with their difference operators[J]. Arch Math,2012,99(3):261-270.

[51] LUO X D,LIN W C.Value sharing results for shifts of meromorphic functions[J]. J Math Anal Appl,2011,377(2):441-449.

[52] QI X G,YANG L Z,LIU K. Uniqueness and periodicity of meromorphic functions concerning the difference operator[J].Computers and Math Appl,2010,60(6):1739-1746.

[53] QI X G,LIU K,YANG L Z. Value sharing results of a meromorphic functionf(z) andf(qz)[J]. Bull Korean Math Soc,2011,48(6):1235-1243.

[54] QI X G,LIU K.Uniqueness and value distribution of differences of entire funtions[J]. J Math Anal Appl,2011,379(1):180-187.

[55] BRüCK R. On entire functions which share one value CM with their first derivate[J]. Results Math,1996,30(1-2):21-24.

[56] CHEN Z X,YI H X. On sharing values of meromorphic functions and their differences[J].Results Math,2013,63(1-2):557-565.

[57] HALBURD R G,KORHONEN R J. Finite-order meromorphic solutions and the discrete Painlevé equations[J].Proc Lond Math Soc,2007,94(2):443-474.

[58] CHEN Z X,SHON K H.Value distribution of meromorphic solutions of certain difference Painlevé equations[J].J Math Anal Appl,2010,364(2):556-566.

[59] CHEN Z X,SHON K H.Some results on difference Riccati equations[J]. Acta Math Sin,2011, 27(6):1091-1100.

[60] CHEN Z X,HUANG Z B,ZHANG R R. On difference equations relating to gamma function[J]. Acta Math Sci:Ser B, 2011,31(4):1377-1382.

[61] SHIMOMURA S. Rational solutions of difference Painleve equations[J]. Tokyo J Math,2012,35(1):85-95.

[62] HALBURD R G,KORHONEN R J. Existence of finite order meromorphic solutions as a detector of integrability in difference equations[J]. Phys D,2006,218(2):191-203.

[63] ISHIZAKI K. On difference Riccati equations and second order linear difference equations[J]. Aequationes Math,2011,81(1-2):185-198.

[64] PANG C W,CHEN Z X. On a conjecture concernng some nonlinear difference equations[J].Bull Malays Math Sci Soc,2013,35(2):765-774.

[65] YANG C C,LAINE I. On analogies between nonlinear difference and differential equations[J]. Proc Japan Acad:Ser A,2010,86(1):10-14.

[66] ZHANG J,LIAO L W. On entire solutions of a certain type of nonlinear differential and difference equations[J]. Taiwanese J Math,2011,15(5):2145-2157.

[67] CHEN Z X.Growth and zeros of meromorphic solution of some linear difference equations[J]. J Math Anal Appl, 2011,373(1):235-241.

[68] HEITTOKANGAS J,KORHONEN R,LAINE I,et al.Complex difference equations of Malmquist type[J].Comput Methods Funct Theory,2001,1(1):27-39.

[69] LAINE I,RIEPPO J,SILVENNOINEN H. Remarks on complex difference equations[J].Comput Methods Funct Theory,2005, 5(1):77-88.

[70] WANG J. Growth and poles of meromorphic solutions of some difference equations[J].J Math Anal Appl,2011,379(1):367-377.

[71] ZHENG X M,CHEN Z X,TU J. Growth of meromorphic solutions of some difference equations[J]. Appl Anal Discrete Math,2010,4(2):309-321.

[72] RITT J F.Transcendental transcendency of certain functions of Poincaré[J].Math Ann,1926,95(1):671-682.

[73] VALIRON G. Fonctons analytiques[M]. Paris:Univ de France, 1954.

[74] ISHIZAKI K,YANAGIIHARA N. Borel and Julia directions of meromorphic Schr?der functions[J]. Math Proc Combridge Philos Soc,2005,139(1):139-147.

[75] ISHIZAKI K,YANAGIIHARA N.Deficiency for meromorphic solutions of Schr?der equations[J]. Complex Var Theory Appl,2004,49(7-9):539-548.

[76] ISHIZAKI K,YANAGIIHARA N. Singular directions of meromorphic solutions of some non-autonomous Schr?der equations[J].Adv Stu Pure Math, 2006,44:155-166.

[77] BERGWEILER W,ISHIZAKI K,YANAGIHARA N. Meromorphic solutions of some functional equations[J]. Methods Appl Anal,1998,5(3):248-258.Erratum: Methods Appl Anal,1999, 6(4): 617-618.

[78] GUNDERSEN G G,HEITTOKANGAS J,LAINE I,et al.Meromorphic solutions of generalized Schr?der equations[J].Aequationes Math,2002,63(1-2):110-135.

[79] HEITTOKANGAS J,LAINE I,RIEPPO J,et al.Meromorphic solutions of some linear functional equations[J]. Aequationes Math,2000,60(1-2):148-166.

[80] HUANG Z B. On q-difference Riccati equations and second-order linear q-difference equations[J].J Complex Anal, 2013: Art 938579, 10 pp.

[81] BERGWEILER W,ISHIZAKI K,YANAGIHARA N.Growth of meromorphic solutions of some functional equations I[J]. Aequationes Math, 2002,63(1-2):140-151.

[82] CHEN B Q,CHEN Z X,LI S. Properties on solutions of some q-difference equations[J]. Acta Math Sin, 2010, 26(10):1877-1886.

[83] CHEN B Q,CHEN Z X.Meromorphic solutions of some q-difference equations[J]. Bull Korean Math,2011,48(6):1303-1314.

[84] LIU K,QI X G. Meromorphic solutions of q-shift difference equations[J]. Ann Polon Math,2011,101(3):215-225.

[85] ZHENG X M,CHEN Z X. Some properties of meromorphic solutions of q-difference equations[J].J Math Anal Appl,2010,361(2):472-480.

[86] ZHENG X M,CHEN Z X. Value distribution of meromorphic solutions of some q-difference equations[J]. J Math Res Exposition, 2011,31(4):698-704.

Keywords： complex difference; difference equations; complexq-difference;q-difference equations; value distribution of meromorphic functions

StudyonComplexDifferencesandDifferenceEquations

CHEN Zongxuan*, HUANG Zhibo

(School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China)

The researches on the complex difference, complexq-difference, difference equations andq-difference equations in recent decades are mainly introduced. These results include the difference analogue of the logarithmic derivative; the difference analogue of Clunie lemma; the difference counterpart of Mohon′ko lemma; the properties on the zeros, fixed-points on complex differences and divided difference of meromorphic functions with small order; the properties on the value distribution of difference polynomials, the properties on the meromorphic solutions of difference Riccati and Painlevé equations; the results onq-differences and meromorphic solutions ofq-difference equations.

2013-07-04

國家自然科學(xué)基金項目(11171119)

*通訊作者:陳宗煊，教授，Email:chzx@vip.sina.com.

1000-5463(2013)06-0026-08

O174.5

A

10.6054/j.jscnun.2013.09.004

【中文責(zé)編：莊曉瓊 英文責(zé)編：肖菁】