DOI: 10.1515/dema-2025-0232 ISSN: 2391-4661
Entire solutions of some Fermat-type partial differential-difference equations in
C
2
${\mathbb{C}}^{2}$
Gorachand Chakraborty, Molla Basir Ahamed, Amiya Gorai Abstract
In this paper, we consider Fermat-type functional equations of the form
F
2
(
z
)
+
G
2
(
z
)
=
exp
(
c
1
z
1
+
c
2
z
2
+
c
0
)
,
$${F}^{2}\left(z\right)+{G}^{2}\left(z\right)=\mathrm{exp}\left({c}_{1}{z}_{1}+{c}_{2}{z}_{2}+{c}_{0}\right),$$
where
F
and
G
are entire functions in
C
2
${\mathbb{C}}^{2}$
and analyze the existence and form of entire solutions to a difference equation and two Fermat-type differential-difference equations involving partial derivatives in
C
2
${\mathbb{C}}^{2}$
, utilizing Nevanlinna theory as our principal method. It is shown that a rank condition – specifically, Rank(
A
) = 2 of the associated coefficient matrix
A
of the corresponding equation – is not sufficient to determine the precise form of the solutions. Under certain conditions related to the coefficients of the equations, we obtain results which contribute to the general theory by extending known one-variable solution frameworks to the setting of several complex variables, thereby refining and generalizing existing results from
C
$\mathbb{C}$
to
C
2
${\mathbb{C}}^{2}$
.