Meromorphic solutions of complex differential-difference equations
In this paper, the authors study the differential-difference equations of the form \[ f(z+c)=e^{P(z)}f^{(k)}(z), \] where \(P(z)\) is a polynomial with \(\deg P=n\geq1\). In fact, they prove that if \(f(z)\) is a finite order entire solution of the above differential-difference equation with a Borel exceptional value b, then \(b = 0\) and the order \(\sigma(f)\) of \(f\) is \(n + 1.\) They also prove that if \(f\) is a transcendental entire solution with finite order of the following differential-difference equation \[ h_n(z)f^{(n)}(z+\eta_n)+\cdots+h_1(z)f'(z+\eta_1)+h_0(z)f(z+\eta_0)=0, \] where \(h_0(z),h_1(z),\cdots, h_n(z)\not\equiv0\) are polynomials, \(\eta_0,\eta_1,\cdots,\eta_n\) are distinct complex constants, then \(\lambda(f-a)\geq \sigma(f)-1\) for every finite value \(a\), where \(\lambda(f-a)\) is the exponent of convergence of zeros of \(f-a\). They also study the following differential-difference equation \[ h_n(z)f^{(n)}(z+\eta_n)+\cdots+h_1(z)f'(z+\eta_1)+h_0(z)f(z+\eta_0)=q(z), \] where \(h_0(z),h_1(z),\cdots, h_n(z)\not\equiv0\) are polynomials, \(\eta_0,\eta_1,\cdots,\eta_n\) are distinct complex constants and \(q(z)\) is a non-zero polynomial.
scientific article; zbMATH DE number 6832339