Existence of nontrivial solutions of linear functional equations
The article deals with the functional equation \[ \sum_{i=1}^n a_i f(b_ix + c_ih) = 0 \qquad (x, h \in {\mathbb C}),\tag{1} \] where \(a_i,b_i,c_i\) are fixed complex numbers, \(f:\;{\mathbb C} \to {\mathbb C}\) is the unknown function. The authors study the conditions under which the equation above has solutions that are generalized polynomials or, in other words, solutions \(f:\;{\mathbb C} \to {\mathbb C}\) such that, for suitable \(n\), \[ \Delta_{h_1} \dots \Delta_{h_{n+1}} f(x) = 0 \qquad (h_1,\dots,h_{n+1} \in {\mathbb C}). \] It is proved that equation (1) has an additive solution \(f\) if and only if \(f\) is an automorphism such that \[ \sum_{i=1}^n a_i f(b_i) = 0 \quad \text{and} \quad \sum_{i=1}^n a_i f(c_i) = 0. \] Further, it is shown that, for every natural \(k\) the following are equivalent: (i) there exists a monomial of degree \(k\) which is a solution of (1); (ii) there exist automorphisms \(\phi_1,\dots,\phi_k\) of \({\mathbb C}\) such that \(\phi_1 \cdots \phi_k\) is a solution of (1); (iii) there exist automorphisms \(\phi_1,\dots,\phi_k\) of \({\mathbb C}\) such that \[ \sum_{i=1}^n a_i \prod_{j \in J} \phi_j(b_i) \prod_{j \notin J} \phi_j(c_i) = 0 \] for every \(J \subseteq \{1,\dots,k\}\). In the special case when all \(a_1,\dots,a_n\) are nonzero, \(b_i = 1\), and \(c_1,\dots,c_n\) are distinct complex numbers, the following are equivalent: (i) there exists a nonconstant solution of (1); (ii) there is a solution of (1) which is an automorphism of \({\mathbb C}\); (iii) \(\sum_{i=1}^n a_i = 0\), and there exist automorphisms \(\phi\) of \({\mathbb C}\) such that \[ \sum_{i=1}^n a_i \phi(c_i) = 0. \] Moreover, in this case, if equation (1) has a solution \(\phi_1 \cdots \phi_k\) with the automorphisms \(\phi_1,\dots,\phi_k\), then every subproduct of these automorphisms is also a solution of (1).
scientific article; zbMATH DE number 6351394