Fifth Painleve Transcendents Which are Analytic at the Origin

Abstract

Abstract. We study special solutions of the fifth Painlevé equation which are analytic around t ¼ 0. We calculate in particular the linear monodromy of those solutions exactly. We also show how those solutions are related to classical solutions in the sense of Umemura.

Critical Behaviours of the Fifth Painlevé Transcendents and the Monodromy Data

“…Since σ 0 ∈ Z, the coefficients c 0 0n (σ 0 ) of both solutions are uniquely determined, and they coincide with the solutions (II) and (III) in [16,Theorem 2], respectively.…”

“…For more general integration constants, a family of solutions near x = 0 expanded into convergent series in spiral domains or sectors was given by the present author [22]. Kaneko and Ohyama [16] presented certain Taylor series solutions around x = 0 such that each corresponding linear system (1.1) is solvable in terms of hypergeometric functions and that the monodromy may be calculated explicitly. Various formal series solutions of (V) were listed by Bryuno and Parusnikova [4,19], who computed them by the method of power geometry.…”

Meromorphic Painlevé transcendents at a fixed singularity

“…We have studied the linear monodromy of meromorphic solutions around t = 0 in 4. Since we use different form from 4, we list up the linear monodromy here.…”

“…We have found three and twelve meromorphic solutions around a fixed singularity for P V and P VI , respectively 4, 5. For \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$P^{\prime }_{\rm III}(D_6)$\end{document} Okumura found two meromorphic solutions around t = 0 and calculated a linear monodromy for one solution 10.…”

“… Abstract We classify solutions of the third Painlevé equation \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$P^{\prime }_{\rm III}$\end{document}, which are meromorphic around the origin, and determine their linear monodromy. Combined with 4, 5, all of meromorphic solutions of the Painlevé equations around fixed singular points of the Briot‐Bouquet type are completely determined. We characterize the linear monodromy of meromorphic solutions for the third, fifth and sixth Painlevé equations. …”