Characterizations of transcendental entire solutions of trinomial partial differential-difference equations in ℂ2#
This study is devoted to exploring the existence and the precise form of finite-order transcendental entire solutions of second-order trinomial partial differential-difference equations L(f)2+2hL(f)f(z1+c1,z2+c2)+f(z1+c1,z2+c2)2=eg(z1,z2)L{(f)}^{2}+2hL(f)f\left({z}_{1}+{c}_{1},{z}_{2}+{c}_{2})+f{\le...
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De Gruyter
2024-11-01
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| Series: | Demonstratio Mathematica |
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| Online Access: | https://doi.org/10.1515/dema-2024-0052 |
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| author | Xu Hong Yan Haldar Goutam |
| author_facet | Xu Hong Yan Haldar Goutam |
| author_sort | Xu Hong Yan |
| collection | DOAJ |
| description | This study is devoted to exploring the existence and the precise form of finite-order transcendental entire solutions of second-order trinomial partial differential-difference equations L(f)2+2hL(f)f(z1+c1,z2+c2)+f(z1+c1,z2+c2)2=eg(z1,z2)L{(f)}^{2}+2hL(f)f\left({z}_{1}+{c}_{1},{z}_{2}+{c}_{2})+f{\left({z}_{1}+{c}_{1},{z}_{2}+{c}_{2})}^{2}={e}^{g\left({z}_{1},{z}_{2})} and L˜(f)2+2hL˜(f)(f(z1+c1,z2+c2)−f(z1,z2))+(f(z1+c1,z2+c2)−f(z1,z2))2=eg(z1,z2),\tilde{L}{(f)}^{2}+2h\tilde{L}(f)(f\left({z}_{1}+{c}_{1},{z}_{2}+{c}_{2})-f\left({z}_{1},{z}_{2}))+{(f\left({z}_{1}+{c}_{1},{z}_{2}+{c}_{2})-f\left({z}_{1},{z}_{2}))}^{2}={e}^{g\left({z}_{1},{z}_{2})}, where L(f)L(f) and L˜(f)\tilde{L}(f) are defined in (2.1) and (2.2), respectively, and g(z)g\left(z) is a polynomial in C2{{\mathbb{C}}}^{2}. Our results are the extensions of some of the previous results of Liu et al. Also, we exhibit a series of examples to explain that the forms of transcendental entire solutions of finite-order in our results are precise. |
| format | Article |
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| language | English |
| publishDate | 2024-11-01 |
| publisher | De Gruyter |
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| series | Demonstratio Mathematica |
| spelling | doaj-art-608c5a8ff2d740d0bdc210bb53b7fc0b2025-08-20T02:23:35ZengDe GruyterDemonstratio Mathematica2391-46612024-11-0157144355110.1515/dema-2024-0052Characterizations of transcendental entire solutions of trinomial partial differential-difference equations in ℂ2#Xu Hong Yan0Haldar Goutam1School of Arts and Sciences, Suqian University, Suqian, Jiangsu 223800, P. R. ChinaDepartment of Mathematics, Ghani Khan Choudhury Institute of Engineering and Technology, Narayanpur, Malda, PIN 732141, West Bengal, IndiaThis study is devoted to exploring the existence and the precise form of finite-order transcendental entire solutions of second-order trinomial partial differential-difference equations L(f)2+2hL(f)f(z1+c1,z2+c2)+f(z1+c1,z2+c2)2=eg(z1,z2)L{(f)}^{2}+2hL(f)f\left({z}_{1}+{c}_{1},{z}_{2}+{c}_{2})+f{\left({z}_{1}+{c}_{1},{z}_{2}+{c}_{2})}^{2}={e}^{g\left({z}_{1},{z}_{2})} and L˜(f)2+2hL˜(f)(f(z1+c1,z2+c2)−f(z1,z2))+(f(z1+c1,z2+c2)−f(z1,z2))2=eg(z1,z2),\tilde{L}{(f)}^{2}+2h\tilde{L}(f)(f\left({z}_{1}+{c}_{1},{z}_{2}+{c}_{2})-f\left({z}_{1},{z}_{2}))+{(f\left({z}_{1}+{c}_{1},{z}_{2}+{c}_{2})-f\left({z}_{1},{z}_{2}))}^{2}={e}^{g\left({z}_{1},{z}_{2})}, where L(f)L(f) and L˜(f)\tilde{L}(f) are defined in (2.1) and (2.2), respectively, and g(z)g\left(z) is a polynomial in C2{{\mathbb{C}}}^{2}. Our results are the extensions of some of the previous results of Liu et al. Also, we exhibit a series of examples to explain that the forms of transcendental entire solutions of finite-order in our results are precise.https://doi.org/10.1515/dema-2024-0052functions of several complex variablesfermat-type equationsentire solutionsnevanlinna theory30d3535m3032w5039a45 |
| spellingShingle | Xu Hong Yan Haldar Goutam Characterizations of transcendental entire solutions of trinomial partial differential-difference equations in ℂ2# Demonstratio Mathematica functions of several complex variables fermat-type equations entire solutions nevanlinna theory 30d35 35m30 32w50 39a45 |
| title | Characterizations of transcendental entire solutions of trinomial partial differential-difference equations in ℂ2# |
| title_full | Characterizations of transcendental entire solutions of trinomial partial differential-difference equations in ℂ2# |
| title_fullStr | Characterizations of transcendental entire solutions of trinomial partial differential-difference equations in ℂ2# |
| title_full_unstemmed | Characterizations of transcendental entire solutions of trinomial partial differential-difference equations in ℂ2# |
| title_short | Characterizations of transcendental entire solutions of trinomial partial differential-difference equations in ℂ2# |
| title_sort | characterizations of transcendental entire solutions of trinomial partial differential difference equations in c2 |
| topic | functions of several complex variables fermat-type equations entire solutions nevanlinna theory 30d35 35m30 32w50 39a45 |
| url | https://doi.org/10.1515/dema-2024-0052 |
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