Shun Shimomura

Meromorphic solutions of difference Painlevé equations

We show that, under certain conditions, some difference Painleve equations have nontrivial meromorphic solutions in the whole complex plane. These meromorphic solutions are obtained by analytic continuation of asymptotic solutions given in sectors of zero opening angle. The existence of these asymptotic solutions is shown by using fundamental matrix solutions of the associated linear systems with double unit characteristic roots and with coefficients expressed by asymptotic series in fractional powers.

Lower estimates for the growth of the fourth and the second painlevé transcendents

Let

Value Distribution of Painlevé Transcendents of The Fifth Kind

w(z)

Value distribution of Painlevé transcendents of the first and the second kind

be an arbitrary transcendental solution of the fourth (respectively, second) Painleve equation. Concerning the frequency of poles in

Oscillation results for {

|z|\le r

n

, it is shown that

}-th order linear differential equations with meromorphic periodic coefficients

n(r,w)\gg r^2

Riccati Differential Equations with Elliptic Coefficients

(respectively,

Pole loci of solutions of a degenerate Garnier system

n(r,w)\gg r^{3/2}

Value distribution of painlevé transcendents of the third kind

), from which the growth estimate