Equi-Distribution of Values for the Third and the Fifth Painlevé Transcendents

Abstract

Abstract. We show equi-distribution properties of values for the third and the fifth Painlevé transcendents in a sectorial domain. For our purpose we define a characteristic function of sectorial domain type by employing value distribution theory in a half plane. Some special cases admit analogues of Borel exceptional values. Similar results are obtained for modified versions of these Painlevé transcendents, which are of infinite growth order.

Frequency of $a$-points for the fifth and the third Painlevé transcendents in a sector

“…(More explanations of these quantities are found in [12] and [13]; and, for the value distribution theory in a half plane or a sector, readers are referred to [4], [5], [15], [16].) THEOREM 2.2.…”

“…For solutions of the fifth and the third Painlevé equations, value distribution properties were studied by Sasaki [6], [7] (see also [11] for their modified versions). From them it follows that, for every a ∈ C ∪ {∞}, the number of a-points in a sector with radius r is at most O(r Λ ), where Λ is some large positive number independent of a; and, for admissible or truncated solutions, we established the equi-distribution property of a-points in a sector [12], [13].…”

Truncated Solutions of the Fifth Painlevé Equation

“…Denote by n l ðr; f Þ, the integration is taken for f ðzÞ along the boundary of Wðy 0 ; l; rÞ outside the circle jz À e iy 0 =2j ¼ 1, since the circle jz À ir 1=l =2j ¼ r 1=l =2 is also expressible as z ¼ r 1=l e if sin f, 0 a f a p. Although the definition of Wðy 0 ; l; rÞ here is slightly di¤erent from that in [6], the quantities m We are ready to state our results on the frequency of a-points. In them, (i) l and c are given numbers satisfying l > 1 and c A Cnf0g, respectively, (ii) a denotes a given value such that a A C U fyg except in Theorems 3.3, (3) and 3.5, (3),…”

“…For each a A C U fyg the first Painlevé transcendents have infinitely many a-points in the whole complex plane, and all the values are equally distributed (see, for example, [3, § 10]). In [6], under a certain condition on the growth order, the counterpart of this fact for the fifth Painlevé transcendents is established in sectors near z ¼ y; for example, oscillatory type solutions on the positive real axis have infinitely many a-points equally distributed in any sectors of enough opening angle.…”