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The Fermat type functional equations $(*)\, f_1^n+f_2^n+\cdots +f_k^n=1$ , where n and k are positive integers, are considered in the complex plane. Our focus is on equations of the form (*) where it is not known whether there exist non-constant solutions in one or more of the following four classes of functions: meromorphic functions, rational functions, entire functions, polynomials. For such equations, we obtain estimates on Nevanlinna functions that transcendental solutions of (*) would have to satisfy, as well as analogous estimates for non-constant rational solutions. As an application, it is shown that transcendental entire solutions of (*) when n = k(k − 1) with k ≥ 3, would have to satisfy a certain differential equation, which is a generalization of the known result when k = 3. Alternative proofs for the known non-existence theorems for entire and polynomial solutions of (*) are given. Moreover, some restrictions on degrees of polynomial solutions are discussed.
The Fermat type functional equations $(*)\, f_1^n+f_2^n+\cdots +f_k^n=1$ , where n and k are positive integers, are considered in the complex plane. Our focus is on equations of the form (*) where it is not known whether there exist non-constant solutions in one or more of the following four classes of functions: meromorphic functions, rational functions, entire functions, polynomials. For such equations, we obtain estimates on Nevanlinna functions that transcendental solutions of (*) would have to satisfy, as well as analogous estimates for non-constant rational solutions. As an application, it is shown that transcendental entire solutions of (*) when n = k(k − 1) with k ≥ 3, would have to satisfy a certain differential equation, which is a generalization of the known result when k = 3. Alternative proofs for the known non-existence theorems for entire and polynomial solutions of (*) are given. Moreover, some restrictions on degrees of polynomial solutions are discussed.
Let n n and m m be two positive integers, and the second-order Fermat-type functional equation f n + ( f ″ ) m ≡ 1 {f}^{n}+{({f}^{^{\prime\prime} })}^{m}\equiv 1 does not have a nonconstant meromorphic solution in the complex plane, except ( n , m ) ∈ { ( 1 , 1 ) , ( 1 , 2 ) , ( 1 , 3 ) , ( 2 , 1 ) , ( 3 , 1 ) } \left(n,m)\in \left\{\left(1,1),\left(1,2),\left(1,3),\left(2,1),\left(3,1)\right\} . The research gives a ready-to-use scheme to study certain Fermat-type functional differential equations in the complex plane by using the Nevanlinna theory, the complex method, and the Weierstrass factorization theorem.
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