DOI: 10.1515/dema-2024-0052 ISSN: 2391-4661

Characterizations of transcendental entire solutions of trinomial partial differential-difference equations in ℂ2#

Hong Yan Xu, Goutam Haldar
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Abstract

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 + 2 h L ( f ) f ( z 1 + c 1 , z 2 + c 2 ) + f ( z 1 + c 1 , z 2 + c 2 ) 2 = e g ( z 1 , z 2 ) 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 + 2 h L ˜ ( f ) ( f ( z 1 + c 1 , z 2 + c 2 ) − f ( z 1 , z 2 ) ) + ( f ( z 1 + c 1 , z 2 + c 2 ) − f ( z 1 , z 2 ) ) 2 = e g ( z 1 , z 2 ) , \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
C 2 {{\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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