Properties of meromorphic solutions of first-order differential-difference equations
For the first-order differential-difference equations of the form A(z)f(z+1)+B(z)f′(z)+C(z)f(z)=F(z),A\left(z)f\left(z+1)+B\left(z)f^{\prime} \left(z)+C\left(z)f\left(z)=F\left(z), where A(z),B(z),C(z)A\left(z),B\left(z),C\left(z), and F(z)F\left(z) are polynomials, the existence, growth, zeros, pol...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
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De Gruyter
2023-12-01
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| Series: | Open Mathematics |
| Subjects: | |
| Online Access: | https://doi.org/10.1515/math-2023-0147 |
| _version_ | 1797405101764116480 |
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| author | Wu Lihao Chen Baoqin Li Sheng |
| author_facet | Wu Lihao Chen Baoqin Li Sheng |
| author_sort | Wu Lihao |
| collection | DOAJ |
| description | For the first-order differential-difference equations of the form A(z)f(z+1)+B(z)f′(z)+C(z)f(z)=F(z),A\left(z)f\left(z+1)+B\left(z)f^{\prime} \left(z)+C\left(z)f\left(z)=F\left(z), where A(z),B(z),C(z)A\left(z),B\left(z),C\left(z), and F(z)F\left(z) are polynomials, the existence, growth, zeros, poles, and fixed points of their nonconstant meromorphic solutions are investigated. It is shown that all nonconstant meromorphic solutions are transcendental when degB(z)<deg{A(z)+C(z)}+1{\rm{\deg }}B\left(z)\lt {\rm{\deg }}\left\{A\left(z)+C\left(z)\right\}+1 and all transcendental solutions are of order at least 1. For the finite-order transcendental solution f(z)f\left(z), the relationship between ρ(f)\rho (f) and max{λ(f),λ(1∕f)}\max \left\{\lambda (f),\lambda \left(1/f)\right\} is discussed. Some examples for sharpness of our results are provided. |
| first_indexed | 2024-03-09T03:05:51Z |
| format | Article |
| id | doaj.art-45830c7feb6d4c30a7cf9aa04fd15bcc |
| institution | Directory Open Access Journal |
| issn | 2391-5455 |
| language | English |
| last_indexed | 2024-03-09T03:05:51Z |
| publishDate | 2023-12-01 |
| publisher | De Gruyter |
| record_format | Article |
| series | Open Mathematics |
| spelling | doaj.art-45830c7feb6d4c30a7cf9aa04fd15bcc2023-12-04T07:59:56ZengDe GruyterOpen Mathematics2391-54552023-12-012118890890210.1515/math-2023-0147Properties of meromorphic solutions of first-order differential-difference equationsWu Lihao0Chen Baoqin1Li Sheng2School of Computer Engineering, Guangzhou City University of Technology, Guangzhou 510800, ChinaSchool of Mathematics and Computer, Guangdong Ocean University, Zhanjiang 524088, ChinaSchool of Mathematics and Computer, Guangdong Ocean University, Zhanjiang 524088, ChinaFor the first-order differential-difference equations of the form A(z)f(z+1)+B(z)f′(z)+C(z)f(z)=F(z),A\left(z)f\left(z+1)+B\left(z)f^{\prime} \left(z)+C\left(z)f\left(z)=F\left(z), where A(z),B(z),C(z)A\left(z),B\left(z),C\left(z), and F(z)F\left(z) are polynomials, the existence, growth, zeros, poles, and fixed points of their nonconstant meromorphic solutions are investigated. It is shown that all nonconstant meromorphic solutions are transcendental when degB(z)<deg{A(z)+C(z)}+1{\rm{\deg }}B\left(z)\lt {\rm{\deg }}\left\{A\left(z)+C\left(z)\right\}+1 and all transcendental solutions are of order at least 1. For the finite-order transcendental solution f(z)f\left(z), the relationship between ρ(f)\rho (f) and max{λ(f),λ(1∕f)}\max \left\{\lambda (f),\lambda \left(1/f)\right\} is discussed. Some examples for sharpness of our results are provided.https://doi.org/10.1515/math-2023-0147differential-difference equationgrowth and zerosmeromorphic solution30d3539a06 |
| spellingShingle | Wu Lihao Chen Baoqin Li Sheng Properties of meromorphic solutions of first-order differential-difference equations Open Mathematics differential-difference equation growth and zeros meromorphic solution 30d35 39a06 |
| title | Properties of meromorphic solutions of first-order differential-difference equations |
| title_full | Properties of meromorphic solutions of first-order differential-difference equations |
| title_fullStr | Properties of meromorphic solutions of first-order differential-difference equations |
| title_full_unstemmed | Properties of meromorphic solutions of first-order differential-difference equations |
| title_short | Properties of meromorphic solutions of first-order differential-difference equations |
| title_sort | properties of meromorphic solutions of first order differential difference equations |
| topic | differential-difference equation growth and zeros meromorphic solution 30d35 39a06 |
| url | https://doi.org/10.1515/math-2023-0147 |
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