一类差分多项式的Picard型定理
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摘要
本文利用值分布论作为工具得到以下结论:令f(z)为一个有限级超越整函数,c_1,c_2为两非零复常数并使得f(z+c_1)≠f(z+c_2),q(z)为非零多项式,则f(z)Δf_(c_1)(z)-q(z)和f(z)Δf_(c_2)(z)-q(z)两者中至少有一个具有无限多个零点.
In this paper, by using the values distribution theory, we obtain that let f(z) be a transcendental entire function of finite order, and c_1, c_2 be two nonzero complex numbers such that f(z+c_1) ≠f(z+c_2) and q(z) be a non-vanishing polynomials, then either f( z)Δfc_1( z)-q( z) or f( z)Δfc_2( z)-q( z) has infinitely many zeros.
In this paper, by using the values distribution theory, we obtain that let f(z) be a transcendental entire function of finite order, and c_1, c_2 be two nonzero complex numbers such that f(z+c_1) ≠f(z+c_2) and q(z) be a non-vanishing polynomials, then either f( z)Δfc_1( z)-q( z) or f( z)Δfc_2( z)-q( z) has infinitely many zeros.
引文
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[2]HAYMAN W K.Picard values of meromorphic functions and their derivatives[J].Ann of Math,1959,70(2):9-42.
[3]LAINE I,YANG Chunchun.Value distribution of difference polynomials[J].Proc Japan Acad Ser A Math Sci,2007,83(8):148-151.
[4]LIU Kai,YANG Lianzhong.Value distribution of the difference operator[J].Arch Math,2009,92(3):270-278.
[5]LATREUCH Z,BELAIDI B.On Picard value problem of some difference polynomials[EB/OL].(2017-03-05)[2017-08-15].http://arxiv.org/pdf/1703.01635.pdf.
[6]CHEN Zongxuan,SHON K,HUANG Zhibo.Complex differences and difference equations[M].Beijing:Science Press,2014.
[7]CHEN Zongxuan.On value distribution of difference polynomials of meromorphic functions[J].Abstr Appl Anal,2011(6):1826-1834.
[8]GUNDERSEN G G.Estimates for the logarithmic derivative of a meromorphic functions.plus similar estimates[J].J London Math Soc,1988,37(2):88-104.
[9]HAYMAN W K.Meromorphic functions[M].Oxford:Oxford Mathematical Monographs Clarendon Press,1964.
[10]CHIANG Y,FENG Shaoji.On the Nevanlinna characteristic of f(z(10)η)and difference equations in the complex plane[J].The Ramanujan J,2008,16:105-129.
[2]HAYMAN W K.Picard values of meromorphic functions and their derivatives[J].Ann of Math,1959,70(2):9-42.
[3]LAINE I,YANG Chunchun.Value distribution of difference polynomials[J].Proc Japan Acad Ser A Math Sci,2007,83(8):148-151.
[4]LIU Kai,YANG Lianzhong.Value distribution of the difference operator[J].Arch Math,2009,92(3):270-278.
[5]LATREUCH Z,BELAIDI B.On Picard value problem of some difference polynomials[EB/OL].(2017-03-05)[2017-08-15].http://arxiv.org/pdf/1703.01635.pdf.
[6]CHEN Zongxuan,SHON K,HUANG Zhibo.Complex differences and difference equations[M].Beijing:Science Press,2014.
[7]CHEN Zongxuan.On value distribution of difference polynomials of meromorphic functions[J].Abstr Appl Anal,2011(6):1826-1834.
[8]GUNDERSEN G G.Estimates for the logarithmic derivative of a meromorphic functions.plus similar estimates[J].J London Math Soc,1988,37(2):88-104.
[9]HAYMAN W K.Meromorphic functions[M].Oxford:Oxford Mathematical Monographs Clarendon Press,1964.
[10]CHIANG Y,FENG Shaoji.On the Nevanlinna characteristic of f(z(10)η)and difference equations in the complex plane[J].The Ramanujan J,2008,16:105-129.