Six-dimensional Painlevé systems and their particular solutions in terms of rigid systems

Abstract

Critical edge behavior in the modified Jacobi ensemble and the Painlevé V transcendentsIn this article, we propose a class of six-dimensional Painlevé systems given as the monodromy preserving deformations of the Fuchsian systems. They are expressed as polynomial Hamiltonian systems of sixth order. We also discuss their particular solutions in terms of the hypergeometric functions defined by fourth order rigid systems. C 2014 AIP Publishing LLC. [http://dx.

Determinant structure for tau-function of holonomic deformation of linear differential equations

“…having no accessory parameter such as Gauß's hypergeometric equation, then the deformation equation of (7.1) possesses a solution written in terms of that of the rigid system (4.1) itself. In this case, our procedure gives a natural interpretation to Suzuki's recent work [22], in which a list of rigid systems or hypergeometric equations appearing in particular solutions to the higher order Painlevé equations is presented.…”

A Higher Order Painlevé System in Two Variables and Extensions of the Appell Hypergeometric Functions <i>F</i>₁, <i>F</i>₂ and <i>F</i>₃

“…; l mþ3 . It is probably true in the case of m ¼ 1, namely the Painlevé system is an ordinary di¤erential one; see Section 5 of [19]. However, in the case of m b 2, the situation is more complicated.…”

“…Appell's F 1 or Lauricella's F D ); see [4,6,8]. In recent years an investigation of hypergeometric solutions of Painlevé systems has been developed greatly; see [6,18,19,20]. We list the obtained hypergeometric solutions of 4th and 6th order Painlevé systems in the following The symbol H l stands for a Painlevé system which is derived from the Fuchsian system with the spectral type l. Note that the Garnier system in two (resp.…”

“…From this point of view several extensions of the Painlevé VI equation have been proposed in [5,6,11,15,19,21].…”

Regular Flat Structure and Generalized Okubo System

“…For generic values of w i (i.e. w i = w j for i = j), solutions to the fourdimensional extended WDVV equation ( 70)-( 72) correspond to generic solutions to the six-dimensional Painlevé equation specified by the spectral type (21,21,21,21,111) which is studied by T. Suzuki [34] and its degeneration family. In the case of w 1 = w 2 , solutions to (70)-( 72) correspond to generic solutions to the Garnier system in two variables and its degeneration family (see e.g.…”