亚纯函数和q-差分多项式分担一个值的唯一性
|
摘要
本文研究超越亚纯函数与其q-差分多项式分担一个值的唯一性理论.设f(Z)为具有有限多个极点的零级超越亚纯函数,对任意n,k∈N,若f~n(z)-Q_1(z),[f(q_1z)f(q_2z)...f(q_nz)]~((k))-Q_2(Z)分担0IM并且f~n(z),f(q_1z)f(q_2z)…f(q_nz)分担0CM,此处q_i(i=1,2,…,n)为非零复常数,Q_1,Q_2为两多项式且满足Q_1Q_2?0.如果n≥k+2,则[f(q_1z)f(q_2z)…f(q_nz)]~((k))≡Q_2(z)f~n(z)/Q_1(Z).
In this paper, we study the uniqueness of a transcendental meromorphic function and its q-difference polynomial when they share a certain value and we get: Let f(z) be a transcendental meromorphic function of zero-order with finitely many poles, and n,k ∈ N,Suppose f~n(z)-Q_1(z),[f(q_1z)f(q_2z)...f(q_nz)]~((k))-Q_2(Z) share OIM and f~n(z),f(q_1z)f(q_2z)…f(q_nz) share OCM where q_i(i=1,2,…,n) are nonzero constants,Q_1,Q_2 are two polynomials with Q_1Q_2?0. If n≥k+2, then we get [f(q_1z)f(q_2z)…f(q_nz)]~((k))≡Q_2(z)f~n(z)/Q_1(Z).
In this paper, we study the uniqueness of a transcendental meromorphic function and its q-difference polynomial when they share a certain value and we get: Let f(z) be a transcendental meromorphic function of zero-order with finitely many poles, and n,k ∈ N,Suppose f~n(z)-Q_1(z),[f(q_1z)f(q_2z)...f(q_nz)]~((k))-Q_2(Z) share OIM and f~n(z),f(q_1z)f(q_2z)…f(q_nz) share OCM where q_i(i=1,2,…,n) are nonzero constants,Q_1,Q_2 are two polynomials with Q_1Q_2?0. If n≥k+2, then we get [f(q_1z)f(q_2z)…f(q_nz)]~((k))≡Q_2(z)f~n(z)/Q_1(Z).
引文
[1] 仪洪勋,杨重骏.亚纯函数唯一性理论[M].北京:科版社学出版,1995.
[2] LU Feng, YI Hongxun. The Brück conjecture and entire functions sharing polynomials with Their k-th derivatives[J]. J Korean Math Soc, 2011, 48(3):499-512.
[3] LU Wenran, LI Qiuying, YANG Chungchun. On the transcendental entire solutions of a class of differential equations[J]. J Korean Math Soc, 2014, 51(5):1281-1289.
[4] MAJUMDER S. A result on a conjecture of W. Lü, Q. Li, C.Yang[J]. J Korean Math Soc, 2016, 53(2):411-421.
[5] MAJUMDER S, SAHA A S. A note on a question of Lü, Li and Yang[J]. Applied Mathematics E-Notes,2018(18):156-166.
[6] YANG Chungchun. On deficiencies of differential polynomials[J]. Mathematische Zeitschrift, 1970, 116(3):197-204.
[7] BARNETT D C, HALBURD R G, KORHONEN R J, et al, Nevanlinna theory for the q-difference operator and meromorphic solutions of q-difference equations[J]. Proc Royal Soc Edinburgh, Sect A, 2007,137(3):457-474.
[8] CHIANG Yikman, FENG Shaoji. On the Nevanlinna characteristic f(z+η)and difference e-quations in complex plane[J]. Ramanujan J Math, 2008, 16(1):105-129.
[9] HEITTOKANGAS J,KORHONEN R, LAINE I,et al. Value sharing results for shifts of meromorphic function,and sufficient conditions for periodicity[J]. J Math Anal Appl, 2009, 355(1):352-363.
[10] HAYMAN W K. Meromorphic functions[J]. Lecture Notes in Mathematics, 1964, 234(369):59-82.
[2] LU Feng, YI Hongxun. The Brück conjecture and entire functions sharing polynomials with Their k-th derivatives[J]. J Korean Math Soc, 2011, 48(3):499-512.
[3] LU Wenran, LI Qiuying, YANG Chungchun. On the transcendental entire solutions of a class of differential equations[J]. J Korean Math Soc, 2014, 51(5):1281-1289.
[4] MAJUMDER S. A result on a conjecture of W. Lü, Q. Li, C.Yang[J]. J Korean Math Soc, 2016, 53(2):411-421.
[5] MAJUMDER S, SAHA A S. A note on a question of Lü, Li and Yang[J]. Applied Mathematics E-Notes,2018(18):156-166.
[6] YANG Chungchun. On deficiencies of differential polynomials[J]. Mathematische Zeitschrift, 1970, 116(3):197-204.
[7] BARNETT D C, HALBURD R G, KORHONEN R J, et al, Nevanlinna theory for the q-difference operator and meromorphic solutions of q-difference equations[J]. Proc Royal Soc Edinburgh, Sect A, 2007,137(3):457-474.
[8] CHIANG Yikman, FENG Shaoji. On the Nevanlinna characteristic f(z+η)and difference e-quations in complex plane[J]. Ramanujan J Math, 2008, 16(1):105-129.
[9] HEITTOKANGAS J,KORHONEN R, LAINE I,et al. Value sharing results for shifts of meromorphic function,and sufficient conditions for periodicity[J]. J Math Anal Appl, 2009, 355(1):352-363.
[10] HAYMAN W K. Meromorphic functions[J]. Lecture Notes in Mathematics, 1964, 234(369):59-82.