Complex oscillation of meromorphic solutions for difference Riccati equation
The authors investigate zeros and \(\alpha\)-points of meromorphic solutions \(f(z)\) for difference Riccati equations \(\displaystyle f (z +1)=\frac{A(z)+\delta f (z)}{\delta-f (z)}\). Let \(\sigma(f)\) be the order of growth of the meromorphic function \(f(z)\), let \(\lambda(f)\) and \(\displaystyle\lambda\left(\frac1f\right)\) be the exponents of convergence of the zeros and the poles of \(f(z)\), respectively. A meromorphic function \(f(z)\) is oscillatory if \(f(z)\) has infinitely many zeros. Some estimates for the exponents of convergence of the zeros and \(\alpha\)-points of \(f(z)\), the shifts \(f(z + n)\), the differences \(\Delta f(z):=f(z + 1)-f(z)\), and the divided differences \(\displaystyle d\Delta(f(z)):=\frac{\Delta f(z)}{f(z)}\) are obtained. Let \(\delta=\pm1\) be a constant and \(A(z)\) be a nonconstant rational function. If there exists a nonconstant rational function \(s(z)\) such that \(A(z)=-s^2(z)\), then every finite order transcendental meromorphic solution \(f(z)\) of the difference Riccati equation, its difference, and divided difference are oscillatory and satisfy \(\displaystyle\lambda(\Delta f(z))=\lambda(d\Delta f(z))=\sigma(f)\).
scientific article; zbMATH DE number 6623117
scientific article