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Volume 41, Issue 1
On Solutions of Differential-Difference Equations in $\mathbb{C}^n$

Ling Yang, Lu Chen & Shimei Zhang

DOI: 10.4208/ata.OA-2024-0044

Anal. Theory Appl., 41 (2025), pp. 52-79.

Published online: 2025-04

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  • Abstract

In this paper, we mainly explore the existence of entire solutions of the quadratic trinomial partial differential-difference equation $$af^2(z)+2\omega f(z)(a_0f(z)+L^{k+s}_{1,2}(f(z)))+b(a_0f(z)+L^{k+s}_{1,2}(f(z)))^2=e^{g(z)}$$
by utilizing Nevanlinna’s theory in several complex variables, where $g(z)$ is entire functions in $\mathbb{C}^n,$ $ω\ne 0$ and $a, b, ω ∈ \mathbb{C}.$ Furthermore, we get the exact froms of solutions of the above differential-difference equation when $ω = 0.$ Our results are generalizations of previous results. In addition, some examples are given to illustrate the accuracy of the results.

  • Keywords

Differential-difference equations, Nevanlinna theory, finite order, entire solutions.

  • AMS Subject Headings

39A45, 30D35, 39A14, 32H30, 35A20

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{ATA-41-52, author = {Yang , LingChen , Lu and Zhang , Shimei}, title = {On Solutions of Differential-Difference Equations in $\mathbb{C}^n$}, journal = {Analysis in Theory and Applications}, year = {2025}, volume = {41}, number = {1}, pages = {52--79}, abstract = {

In this paper, we mainly explore the existence of entire solutions of the quadratic trinomial partial differential-difference equation $$af^2(z)+2\omega f(z)(a_0f(z)+L^{k+s}_{1,2}(f(z)))+b(a_0f(z)+L^{k+s}_{1,2}(f(z)))^2=e^{g(z)}$$
by utilizing Nevanlinna’s theory in several complex variables, where $g(z)$ is entire functions in $\mathbb{C}^n,$ $ω\ne 0$ and $a, b, ω ∈ \mathbb{C}.$ Furthermore, we get the exact froms of solutions of the above differential-difference equation when $ω = 0.$ Our results are generalizations of previous results. In addition, some examples are given to illustrate the accuracy of the results.

}, issn = {1573-8175}, doi = {https://doi.org/10.4208/ata.OA-2024-0044}, url = {http://global-sci.org/intro/article_detail/ata/23958.html} }
TY - JOUR T1 - On Solutions of Differential-Difference Equations in $\mathbb{C}^n$ AU - Yang , Ling AU - Chen , Lu AU - Zhang , Shimei JO - Analysis in Theory and Applications VL - 1 SP - 52 EP - 79 PY - 2025 DA - 2025/04 SN - 41 DO - http://doi.org/10.4208/ata.OA-2024-0044 UR - https://global-sci.org/intro/article_detail/ata/23958.html KW - Differential-difference equations, Nevanlinna theory, finite order, entire solutions. AB -

In this paper, we mainly explore the existence of entire solutions of the quadratic trinomial partial differential-difference equation $$af^2(z)+2\omega f(z)(a_0f(z)+L^{k+s}_{1,2}(f(z)))+b(a_0f(z)+L^{k+s}_{1,2}(f(z)))^2=e^{g(z)}$$
by utilizing Nevanlinna’s theory in several complex variables, where $g(z)$ is entire functions in $\mathbb{C}^n,$ $ω\ne 0$ and $a, b, ω ∈ \mathbb{C}.$ Furthermore, we get the exact froms of solutions of the above differential-difference equation when $ω = 0.$ Our results are generalizations of previous results. In addition, some examples are given to illustrate the accuracy of the results.

Yang , LingChen , Lu and Zhang , Shimei. (2025). On Solutions of Differential-Difference Equations in $\mathbb{C}^n$. Analysis in Theory and Applications. 41 (1). 52-79. doi:10.4208/ata.OA-2024-0044
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