#
Solutions for Several Quadratic Trinomial Difference Equations and Partial Differential Difference Equations in ${\mathbb{C}}^{2}$

^{1}

^{2}

^{3}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

**Theorem**

**1**

**Theorem**

**2**

**Theorem**

**3**

- (i)
- $u=f({c}_{1}{z}_{1}+{c}_{2}{z}_{2})$; or
- (ii)
- $u={\varphi}_{1}({z}_{1}+i{z}_{2})+{\varphi}_{2}({z}_{1}-i{z}_{2})$,

**Theorem**

**4**

**Question**

**1.**

## 2. Results and Examples

**Theorem**

**5.**

**Example**

**1.**

**Remark**

**1.**

**Example**

**2.**

**Theorem**

**6.**

**Theorem**

**7.**

**Example**

**3.**

**Example**

**4.**

**Corollary**

**1.**

**Theorem**

**8.**

**Example**

**5.**

**Example**

**6.**

**Example**

**7.**

**Remark**

**2.**

**Theorem**

**9.**

**Example**

**8.**

**Example**

**9.**

**Example**

**10.**

**Remark**

**3.**

## 3. Conclusions and Discussion

## 4. Some Lemmas

**Lemma**

**1**

**Lemma**

**2**

**Remark**

**4.**

**Lemma**

**3**

**Lemma**

**4**

**Remark**

**5.**

## 5. The Proof of Theorem 5

**Proof.**

**Case 1.**Suppose that ${\gamma}_{1}(z+c)-{\gamma}_{2}\left(z\right)$ is a constant. Let ${\gamma}_{1}(z+c)-{\gamma}_{2}\left(z\right)=\kappa $, $\kappa \in \mathbb{C}$. In view of (20), it follows that ${\gamma}_{1}\left(z\right)-{\gamma}_{2}\left(z\right)=2p\left(z\right)$. Substituting these into (23), we have

**Case 2.**Suppose that ${\gamma}_{1}(z+c)-{\gamma}_{2}\left(z\right)$ is not a constant. Since ${\gamma}_{1}\left(z\right),{\gamma}_{2}\left(z\right)$ are polynomials, by applying Lemma 4 for (23), it follows that

## 6. Proofs of Theorems 6 and 7

#### 6.1. The Proof of Theorem 6

#### 6.2. The Proof of Theorem 7

## 7. Proofs of Theorems 8 and 9

#### 7.1. The Proof of Theorem 8

**Case 1**. Suppose that $p\left(z\right)$ is a constant; then, it follows that ${\gamma}_{1}(z+c)-{\gamma}_{2}(z+c)=2p(z+c)$ is a constant. Denote $\xi ={e}^{p}$. In view of (20) and (42), it follows that

**Case 2**. Suppose that $p\left(z\right)$ is not a constant. Then, we have that $\frac{\partial {\gamma}_{1}}{\partial {z}_{1}}$ and $\frac{\partial {\gamma}_{2}}{\partial {z}_{1}}$ cannot be equal to 0 at the same time. Otherwise, it yields that ${\gamma}_{1}(z+c)-{\gamma}_{2}(z+c)$ is a constant, which is a contradiction. If $\frac{\partial {\gamma}_{1}}{\partial {z}_{1}}\equiv 0$ and $\frac{\partial {\gamma}_{2}}{\partial {z}_{1}}\not\equiv 0$, it thus follows from (43) that

**Subcase 2.1**. If $\frac{{A}_{1}}{{A}_{2}}\frac{\partial {\gamma}_{1}}{\partial {z}_{1}}{e}^{{\gamma}_{1}\left(z\right)-{\gamma}_{1}(z+c)}\equiv 1$, it yields that ${\gamma}_{1}\left(z\right)-{\gamma}_{1}(z+c)$ is a constant. This implies that ${\gamma}_{1}\left(z\right)$ is a linear form of ${\gamma}_{1}\left(z\right)={L}_{1}\left(z\right)+{H}_{1}\left(s\right)$, where ${L}_{1}\left(z\right)={a}_{11}{z}_{1}+{a}_{12}{z}_{2}$, ${a}_{11},{a}_{12}$ are constants, and ${H}_{1}\left(s\right)$ is a polynomial in $s={c}_{2}{z}_{1}-{c}_{1}{z}_{2}$. Thus, it follows that

**Subcase 2.2**. If $\frac{\partial {\gamma}_{2}}{\partial {z}_{1}}{e}^{{\gamma}_{2}\left(z\right)-{\gamma}_{1}(z+c)}\equiv 1$, this means that ${\gamma}_{2}\left(z\right)-{\gamma}_{1}(z+c)$ is a constant, without loss of generalization, denote

#### 7.2. The Proof of Theorem 9

**Case 1**. Suppose that $p\left(z\right)$ is a constant. In view of ${\gamma}_{1}(z+c)-{\gamma}_{2}(z+c)=2p(z+c)$, it follows that ${\gamma}_{1}(z+c)-{\gamma}_{2}(z+c)$ is a constant. Denote $\xi ={e}^{p}$. In view of (20) and (55), it follows that

**Case 2**. Suppose that $p\left(z\right)$ is not a constant. Then, we have that $\frac{\partial {\gamma}_{1}}{\partial {z}_{1}}+\frac{\partial {\gamma}_{1}}{\partial {z}_{2}}$ and $\frac{\partial {\gamma}_{2}}{\partial {z}_{1}}+\frac{\partial {\gamma}_{2}}{\partial {z}_{2}}$ cannot be equal to 0 at the same time. Otherwise, it yields that ${\gamma}_{1}(z+c)-{\gamma}_{2}(z+c)$ is a constant, which is a contradiction. If $\frac{\partial {\gamma}_{1}}{\partial {z}_{1}}+\frac{\partial {\gamma}_{1}}{\partial {z}_{2}}\equiv 0$ and $\frac{\partial {\gamma}_{2}}{\partial {z}_{1}}+\frac{\partial {\gamma}_{2}}{\partial {z}_{2}}\not\equiv 0$, it thus follows from (55) that

**Subcase 2.1**. If $\frac{{A}_{1}}{{A}_{2}}\left(\right)open="("\; close=")">\frac{\partial {\gamma}_{1}}{\partial {z}_{1}}+\frac{\partial {\gamma}_{1}}{\partial {z}_{2}}$, it yields that ${\gamma}_{1}\left(z\right)-{\gamma}_{1}(z+c)$ is a constant. This implies that ${\gamma}_{1}\left(z\right)$ is a linear form of ${\gamma}_{1}\left(z\right)={L}_{1}\left(z\right)+{H}_{1}\left(s\right)$, where ${L}_{1}\left(z\right)={a}_{11}{z}_{1}+{a}_{12}{z}_{2}$, ${a}_{11},{a}_{12}$ are constants, and ${H}_{1}\left(s\right)$ is a polynomial in $s={c}_{2}{z}_{1}-{c}_{1}{z}_{2}$. Thus, it follows that

**Subcase 2.2**. If $(\frac{\partial {\gamma}_{2}}{\partial {z}_{1}}+\frac{\partial {\gamma}_{2}}{\partial {z}_{2}}){e}^{{\gamma}_{2}\left(z\right)-{\gamma}_{1}(z+c)}\equiv 1$, similar to the argument as the proof of Subcase 2.2 in Theorem 8, we have $p\left(z\right)+p(z+c)=\frac{1}{2}({\mu}_{2}-{\mu}_{1}),$ which contradicts with the assumption that ${\gamma}_{1}(z+c)-{\gamma}_{2}(z+c)=2p(z+c)$ is not a constant. Thus, we get the conclusions of Theorem 9 (ii) from (63)–(65).

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Xu, L.; Cao, T.B. Solutions of complex Fermat-type partial difference and differential-difference equations. Mediterr. J. Math.
**2018**, 15, 1–14, Correction in:**2020**, 17, 1–4.. [Google Scholar] [CrossRef] [Green Version] - Gross, F. On the equation f
^{n}+g^{n}= 1. Bull. Am. Math. Soc.**1966**, 72, 86–88. [Google Scholar] [CrossRef] [Green Version] - Montel, P. Lecons sur les Familles Normales de Fonctions Analytiques et Leurs Applications; Gauthier-Villars: Paris, France, 1927; pp. 135–136. [Google Scholar]
- Pólya, G. On an integral function of an integral function. J. Lond. Math. Soc.
**1926**, 1, 12–15. [Google Scholar] [CrossRef] - Cao, T.; Korhonen, R. A new version of the second main theorem for meromorphic mappings intersecting hyperplanes in several complex variables. J. Math. Anal. Appl.
**2016**, 444, 1114–1132. [Google Scholar] [CrossRef] [Green Version] - 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. [Google Scholar] [CrossRef] - Halburd, R.G.; Korhonen, R. Finite-order meromorphic solutions and the discrete Painlevé equations. Proc. Lond. Math. Soc.
**2007**, 94, 443–474. [Google Scholar] [CrossRef] [Green Version] - Halburd, R.G.; Korhonen, R.J. Nevanlinna theory for the difference operator. Ann. Acad. Sci. Fenn.
**2006**, 31, 463–478. [Google Scholar] - Ronkin, L.I. Introduction to the Theory of Entire Functions of Several Variables; Nauka: Moscow, Russian, 1971. [Google Scholar]
- Stoll, W. Holomorphic Functions of Finite Order in Several Complex Variables; American Mathematical Society: Providence, RI, USA, 1974. [Google Scholar]
- Vitter, A. The lemma of the logarithmic derivative in several complex variables. Duke Math. J.
**1977**, 44, 89–104. [Google Scholar] [CrossRef] - Cao, T.B.; Xu, L. Logarithmic difference lemma in several complex variables and partial difference equations. Ann. Mat.
**2020**, 199, 767–794. [Google Scholar] [CrossRef] [Green Version] - Khavinson, D. A note on entire solutions of the eiconal equation. Am. Math. Mon.
**1995**, 102, 159–161. [Google Scholar] [CrossRef] - Li, B.Q. On entire solutions of Fermat type partial differential equations. Int. J. Math.
**2004**, 15, 473–485. [Google Scholar] [CrossRef] - Li, B.Q. Entire solutions of certain partial differential equations and factorization of partial derivatives. Tran. Amer. Math. Soc.
**2004**, 357, 3169–3177. [Google Scholar] [CrossRef] [Green Version] - Li, B.Q. Entire solutions of eiconal type equations. Arch. Math.
**2007**, 89, 350–357. [Google Scholar] [CrossRef] - Lü, F.; Lü, W.; Li, C.; Xu, J. Growth and uniqueness related to complex differential and difference equations. Results Math.
**2019**, 74, 30. [Google Scholar] [CrossRef] - Xu, H.Y.; Liu, S.Y.; Li, Q.P. Entire solutions for several systems of nonlinear difference and partial differentialdifference equations of fermat-type. J. Math. Anal. Appl.
**2020**, 483, 123641. [Google Scholar] [CrossRef] - Zhang, J. On some special difference equations of Malmquist type. Bull. Korean Math. Soc.
**2018**, 55, 51–61. [Google Scholar] - Saleeby, E.G. Entire and meromorphic solutions of Fermat type partial differential equations. Analysis
**1999**, 19, 69–376. [Google Scholar] [CrossRef] - Saleeby, E.G. On entire and meromorphic solutions of λu
^{k}+${\sum}_{i=1}^{n}{u}_{Zi}^{m}$=1. Complex Var. Theory Appl. Int. J.**2004**, 49, 101–107. [Google Scholar] - Li, B.Q. Entire solutions of (${u}_{{z}_{1}}$)
^{m}+(${u}_{{z}_{2}}$)^{n}=e^{g}. Nagoya Math. J.**2005**, 178, 151–162. [Google Scholar] [CrossRef] [Green Version] - Saleeby, E.G. On complex analytic solutions of certain trinomial functional and partial differential equations. Aequat. Math.
**2013**, 85, 553–562. [Google Scholar] [CrossRef] - Liu, K.; Yang, L.Z. A note on meromorphic solutions of Fermat types equations. An. Ştiinţ. Univ. Al. I Cuza Iaşi Mat. (N. S.)
**2016**, 1, 317–325. [Google Scholar] - Berenstein, C.A.; Chang, D.C.; Li, B.Q. On the shared values of entire functions and their partial differential polynomials in ${\mathbb{C}}^{n}$. Forum Math.
**1996**, 8, 379–396. [Google Scholar] [CrossRef] - Hu, P.-C.; Li, P.; Yang, C.-C. Unicity of Meromorphic Mappings, Advances in Complex Analysis and its Applications; Kluwer Academic: Dordrecht, The Netherlands; Boston, MA, USA; London, UK, 2003; Volume 1. [Google Scholar]

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Li, H.; Xu, H.
Solutions for Several Quadratic Trinomial Difference Equations and Partial Differential Difference Equations in *Axioms* **2021**, *10*, 126.
https://doi.org/10.3390/axioms10020126

**AMA Style**

Li H, Xu H.
Solutions for Several Quadratic Trinomial Difference Equations and Partial Differential Difference Equations in *Axioms*. 2021; 10(2):126.
https://doi.org/10.3390/axioms10020126

**Chicago/Turabian Style**

Li, Hong, and Hongyan Xu.
2021. "Solutions for Several Quadratic Trinomial Difference Equations and Partial Differential Difference Equations in *Axioms* 10, no. 2: 126.
https://doi.org/10.3390/axioms10020126