a study of special functions defined by functional equations

1. 利用积分表示，给出了合流超几何函数的渐近性质。应用这一结果，我们还检验了一个合流Pochhammer方程解的整体行为。考察了第三类和第五类painlevel超越的值分布。对于第一类、第二类和第四类的Painleve超越，我们分别用两种不同的方法证明了生长序的有限性。对于这些painlevel超越，我们考察了小函数的不足。3 .我们证明了一类退化Garnier系统的Painleve性质，该系统是第一个Painleve方程的两变量版本，并且证明了它的每一个解的奇异轨迹都是用四阶非线性常微分方程的解表示的解析集。对于具有双周期亚纯系数的线性微分方程，研究了其亚纯解的值分布和生长顺序。对于椭圆型系数Riccati微分方程，证明了在一定条件下，每一个周期解都是双周期的，并得到了双周期解的表达式。

1. We clarified the asymptotic behavior of a confluent hypergeometric function by using an integral representation. Applying the result we also examined the global behavior of solutions of a confluent Pochhammer equation.2. We examined the value distribution of Painleve transcendents of the third and the fifth kind. For Painleve transcendents of the first, the second and the fourth kind, we proved the finiteness of the growth order by two different methods. For these Painleve transcendents we examined the deficiency of small functions.3. We proved the Painleve property of a degenerate Garnier system which is a two-variable version of the first Painleve equation, and also proved that, for every solution of it, the singular loci are analytic sets expressed in terms of solutions of a fourth-order nonlinear ordinary differential equation.4. For linear differential equations with doubly periodic meromorphic coefficients, we examined the value distribution and the growth order of meromorphic solutions. For Riccati differential equations with elliptic coefficients, we proved that, under certain conditions, every periodic solution is doubly periodic, and obtained expressions of such solutions.