亚纯函数与其q阶差分分担公共值问题的研究

数学物理学报 ›› 2014, Vol. 34 ›› Issue (6): 1372-1380.

亚纯函数与其q阶差分分担公共值问题的研究

祁晓光¹, 杨连中²

1.济南大学数学科学学院济南 250022|2.山东大学数学学院济南 250100

收稿日期:2013-07-10 修回日期:2014-06-23 出版日期:2014-12-25 发布日期:2014-12-25
基金资助:
国家自然科学基金(11301220, 11371225)、数学天元基金(11226094)、山东省自然科学基金(ZR2012AQ020, ZR2010AM030)和济南大学博士基金(XBS1211)资助.

Value Sharing for a Meromorphic Function with Its q-Shift Difference

QI Xiao-Guang¹, YANG Lian-Zhong²

1.University of Jinan, School of Mathematics, Jinan |250022|2.Shandong University, School of Mathematics, |Jinan |250100

Received:2013-07-10 Revised:2014-06-23 Online:2014-12-25 Published:2014-12-25
Supported by:
国家自然科学基金(11301220, 11371225)、数学天元基金(11226094)、山东省自然科学基金(ZR2012AQ020, ZR2010AM030)和济南大学博士基金(XBS1211)资助.

摘要/Abstract

摘要：

该文主要探讨了亚纯函数f(z) 与其q 阶差分算子Δ_{q, c}f 分担公共值的问题, 是文献[15]研究内容的延续. 例如, 得到零级亚纯函数f(z) 与 Δ_{q, c}f = f(qz+c)-f(z)分担四个公共值IM, 则有f(z)=Δ_q, _cf 成立. 另外, 当函数的级不为整数或无穷时, 同样得到了f(z) 与Δ_q, _cf 的相关分担结果.

关键词: q 阶差分, 亚纯函数, Nevanlinna 理论, 分担值

Abstract:

This research is a continuation of a recent paper [15]. Shared value problems related to a meromorphic function
f(z) and its q-difference operator Δ_q, _cf are studied. It is shown, for instance, that if f(z) is of zero order and shares four values IM with its q-difference operator Δ_q, _cf = f(qz+c)-f(z), then f(z)=Δ_{q, c}f. Moreover, we also consider problems on sharing values of f(z) and Δ_q, _cf when their orders are not an integer or infinite.

Key words: q-Shift difference, Meromorphic functions, Nevanlinna theory, Sharing value

中图分类号:

30D35

祁晓光, 杨连中. 亚纯函数与其q阶差分分担公共值问题的研究[J]. 数学物理学报, 2014, 34(6): 1372-1380.

QI Xiao-Guang, YANG Lian-Zhong. Value Sharing for a Meromorphic Function with Its q-Shift Difference[J]. Acta mathematica scientia,Series A, 2014, 34(6): 1372-1380.

参考文献

[1] Chen Z X, Yi H X. On sharing values of meromorphic functions and their differences. Results Math, 2013, 63: 557--565

[2] Chiang Y M, Feng S J. On the Nevanlinna characteristic of f(z+η) and difference equations in the complex plane. Ramanujan J, 2008, 16: 105--129

[3] Gundersen G G. Meromorphic functions that share three or four values. J London Math Soc, 1979, 20: 457--466

[4] Gundersen G G. Meromorphic functions that share four values. Trans Amer Math Soc, 1983, 277: 545--567

[5] Gundersen G G. Meromorphic functions that share three values IM and a fourth value CM. Complex Variables Theory Appl, 1992, 20: 99--106

[6] Hayman W. Meromorphic Functions. Oxford: Clarendon Press, 1964

[7] Halburd R G, Korhonen R J. Nevanlinna theory for the difference operator. Ann Acad Sci Fenn Math, 2006, 31: 463--478

[8] Heittokangas J, Korhonen R, Laine I, Rieppo J, Zhang J L. Value sharing results for shifts of meromorphic functions, and sufficient conditions for periodicity. J Math Anal Appl, 2009, 355: 352--363

[9] Hua X H. A unicity theorem for entire functions. Bull London Math Soc, 1990, 22: 457--462

[10] Korhonen R. An extension of Picard's theorem for meromorphic functions of small hyper-order. J Math Anal Appl, 2009, 357: 244--253

[11] Li Y H. Uniqueness theorems for meromorphic functions of order less than one. Northeast Math, 2000, 16: 411--416

[12] Li S, Gao Z S. Entire functions sharing one or two fiite values CM with their shifts or difference operators. Arch Math, 2011, 97: 475--483

[13] Liu K, Qi X G. Meromorphic solutions of q-shift difference equations. Ann Polon Math, 2011, 101: 215--225

[14] Nevanlinna R. Le th\'{e}or\`{e}me de Picard-Borel et la th\'{e}orie des fonctions m\'{e}romorphes. Gauthiers-Villars, Paris, 1929

[15] Qi X G, Yang L Z, Liu Y. Nevanlinna theory for the f(qz+c) and its applications. Acta Math Sci, 2013, 33A(5): 819--828

[16] Shibazaki K. Unicity theorem for entire functions of finite order. Mem Nat Defense Acad Japan, 1981, 21: 67--71

[17] Yang C C. On two entire functions which together with their first derivative have the same zeros. J Math Anal Appl, 1976, 56: 1--6

[18] Yang C C, Yi H X. Uniqueness Theory of Meromorphic Functions. Dordrecht: Kluwer Academic Publishers, 2003

[19] Yang L Z, Zhang J L. Non-existence of meromorphic solution of a Fermat type functional equation. Aequations Math, 2008, 76: 140--150

[20] Yi H X. Uniqueness theorems for meromorphic functions concerning small functions. Indian J Pure Appl Math, 2001, 32: 903--914