2025/04/03 by Liu, Xiong-Feng, Wen, Zhi-Tao, Zhu, Can-Xin
#Complex Variables (math.CV) #FOS: Mathematics
paper · doi:10.48550/arxiv.2504.02243
In this paper, we study the growth of transcendental entire solutions of linear difference equations Pm(z)Δmf(z)+⋯+P1(z)Δf(z)+P0(z)f(z)=0, where Pj(z) are polynomials for j=0,…,m. At first, we reveal type of binomial series in terms of its coefficients. Second, we give a list of all possible orders, which are less than 1, and types of transcendental entire solutions of linear difference equations (+). In particular, we give so far the best precise growth estimate of transcendental entire solutions of order less than 1 of (+), which improves results in [3, 4], [5], [7]. Third, for any given rational number ρ∈(0,1) and real number σ∈(0,∞), we can construct a linear difference equation with polynomial coefficients which has a transcendental entire solution of order ρ and type σ. At last, some examples are illustrated for our main theorem.