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On analogies between nonlinear difference and differential equations. (English) Zbl 1207.34118

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 n2 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.

34M05Entire and meromorphic solutions (ODE)
30D35Distribution of values (one complex variable); Nevanlinna theory