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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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
De Gruyter
2024-11-01
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| Series: | Demonstratio Mathematica |
| Subjects: | |
| Online Access: | https://doi.org/10.1515/dema-2024-0052 |
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| Summary: | 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. |
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| ISSN: | 2391-4661 |