Growth and zeros of meromorphic solution of some linear difference equations
The article deals with the growth of meromorphic solutions and the growth of their zeros for linear difference equation \[ P_n(z)f(z + n) + \dots + P_1(z)f(z + 1) + P_0(z)f(z) = F(z)\tag{1} \] where \(F(z),P_0(z),P_1(z),\dots,P_n(z)\) are polynomials with \(FP_0P_n \not\equiv 0\) and \[ \deg (P_n + \dots + P_1 + P_0) = \max \;\{\deg P_j:\;j = 0,1,\dots,n\} \geq 1.\tag{2} \] The following three theorems are proved: (i) If (2) holds then every finite order transcendental meromorphic solution \(f(z)\) to (1) satisfies \(\sigma(f) \geq 1\) and \(\sigma(f) = \lambda(f)\); (ii) If (2) holds and \(F(z) \equiv 0\) then every finite order transcendental meromorphic solution \(f(z)\) to (1) satisfies \(\sigma(f) \geq 1\), takes every non-zero value \(a \in {\mathbb C}\) infinitely often and \(\sigma(f) = \lambda(f - a)\); (iii) Without assumption (2) every meromorphic solution \(f(z)\) to (1) with infinitely many poles satisfies \(\sigma(f) \geq 1\). In these results \(\sigma(f)\) is the order of growth of meromorphic function \(f(z)\) and \(\lambda(f)\) the exponent of the growth of zeros of \(f(z)\). In the article there are some simple illustrative examples.
scientific article; zbMATH DE number 5804455
scientific article