## Rocky Mountain Journal of Mathematics

### Nevanlinna uniqueness of linear difference polynomials

#### Abstract

In this paper, we investigate shared value problems related to an entire function $f(z)$ of hyper-order less than one and its linear difference polynomial $L(f)=\sum _{i=1}^{k}a_{i}f(z+c_{i})$, where $a_{i}, c_{i}\in \mathbb {C}$. We give sufficient conditions in terms of weighted value sharing and truncated deficiencies, which imply that $L(f)\equiv f$.

#### Article information

Source
Rocky Mountain J. Math., Volume 47, Number 3 (2017), 905-926.

Dates
First available in Project Euclid: 24 June 2017

https://projecteuclid.org/euclid.rmjm/1498269816

Digital Object Identifier
doi:10.1216/RMJ-2017-47-3-905

Mathematical Reviews number (MathSciNet)
MR3682154

Zentralblatt MATH identifier
1372.30020

#### Citation

Li, Nan; Korhonen, Risto; Yang, Lianzhong. Nevanlinna uniqueness of linear difference polynomials. Rocky Mountain J. Math. 47 (2017), no. 3, 905--926. doi:10.1216/RMJ-2017-47-3-905. https://projecteuclid.org/euclid.rmjm/1498269816

#### References

• Z.X. Chen and H.X. Yi, On sharing values of meromorphic functions and their differences, Result. Math. 63 (2013), 557–565.
• Y.M. Chiang and S.J. Feng, On the Nevanlinna characteristic of $f(z+\eta)$ and difference equations in the complex plane, Ramanujan J. 16 (2008), 105–129.
• R.G. Halburd and R.J. Korhonen, Difference analogue of the lemma on the logarithmic derivative with applications to difference equations, J. Math. Anal. Appl. 314 (2006), 477–487.
• ––––, Nevanlinna theory for the difference operator, Ann. Acad. Sci. Math. 31 (2006), 463–478.
• ––––, Meromorphic solutions of difference equations, integrability and the discrete Painlevé equations, J. Phys. Math. Th. 40 (2007), R1–R38.
• R.G. Halburd, R.J. Korhonen and K. Tohge, Holomorphic curves with shift-invariant hyperplane preimages, Trans. Amer. Math. Soc. 366 (2014), 4267–4298.
• W.K. Hayman, Meromorphic functions, Oxford Math. Mono., Clarendon Press, Oxford, 1964.
• J. Heittokangas, R. Korhonen, I. Laine and J. Rieppo, Uniqueness of meromorphic functions sharing values with their shifts, Complex Var. Ellip. Eq. 56 (2011), 81–92.
• J. Heittokangas, R. Korhonen, I. Laine, J. Rieppo and J.L. Zhang, Value sharing results for shifts of meromorphic function, and sufficient conditions for periodicity, J. Math. Anal. Appl. 355 (2009), 352–363.
• I. Laine, Nevanlinna theory and complex differential equations, de Gruyter Stud. Math. 15, de Gruyter, Berlin, 1993.
• K. Liu, Meromorphic functions sharing a set with applications to difference equations, J. Math. Anal. Appl. 359 (2009), 384–393.
• X.G. Qi, L.Z. Yang and K. Liu, Uniqueness and periodicity of meromorphic functions concerning difference operator, Comp. Math. Appl. 60 (2010), 1739–1746.
• H.X. Yi, Uniqueness of meromorphic functions and a question of C.C.Yang, Complex Var. 14 (1990), 169–176.
• ––––, Uniqueness theorems for meromorphic functions whose $n$-th derivatives share the same $1$-points, Complex Var. 34 (1997), 421–436.
• H.X. Yi and C.C. Yang, Uniqueness theory of meromorphic functions, Kluwer Academic Publishers, Dordrecht, 2003.