On analogies between nonlinear difference and differential equations
The authors study some similarities between results on the existence and uniqueness of finite order entire solutions of nonlinear differential equations and difference-differential equations of the form \[ f^n + L(z, f) = h, \] where \(n \geq 2\) is an integer, \(h\) is a given non-vanishing meromorphic function of finite order, and \(L (z, f)\) is a linear difference-differential polynomials, with small meromorphic functions as coefficients. The authors prove for example the following theorem: Theorem 1. Let \(p, q\) be polynomials. Then the nonlinear difference equation \[ f^2 (z) + q (z) f (z +1)=p(z) \] has no transcendental entire solutions of finite order.
scientific article; zbMATH DE number 5690867
scientific article