差分Painlevé方程组解的存在性和增长级

The existence and growth of solutions for difference Painlevéequations

研究差分Painlevé方程组{x(z+1)+x(z)=(a_(1)z+b_(1))y(z)+c_(1)y(z)^(2)/y(z)^(2)-1 y(z)+y(z-1)=(a_(2)z+b_(2))x(z)+c_(2)x(z)^(2)/x(z)^(2)-1的有理函数解的存在性,并给出例子说明我们结果是精确的。进而,我们还研究了此方程组的超越亚纯解的增长级下界的精确估计:在一些条件下,其超越亚纯解(x(z),y(z))的增长级满足ρ(x)=ρ(y)≥1。

In this paper,we mainly study the existence of rational solutions of difference Painlevéequations {x(z+1)+x(z)=(a_(1)z+b_(1))y(z)+c_(1)y(z)^(2)/y(z)^(2)-1 y(z)+y(z-1)=(a_(2)z+b_(2))x(z)+c_(2)x(z)^(2)/x(z)^(2)-1,obtain the condition that there is no rational solution to the system of equations and a counter example is given.Furthermore,we also study the exact estimation of the growth order of the transcendental meromorphic solution of the system:under some conditions,the growth order of the transcendental meromorphic solution(x(z),y(z))satisfiesρ(x)=ρ(y)≥1.