Certain functional identities on division rings

We study the functional identity G(x)f(x) = H(x) on a division ring D, where f: D. Dis an additive map and G(X) not equal 0, H(X) are generalized polynomials in the variable Xwith coefficients in D. Precisely, it is proved that either Dis finite-dimensional over its center or fis an elementary operator. Applying the result and its consequences, we prove that if Dis a noncommutative division ring of characteristic not 2, then the only solution of additive maps f, gon Dsatisfying the identity f(x) = xng(x-1) with n not equal 2a positive integer is the trivial case, that is, f= 0and g= 0. This extends Catalano and Merchan's result in 2023 to get a complete solution.