Fuchs’ solution of Painlevé VI equation by means of Jacobi’s last multiplier

2007 ◽  
Vol 48 (1) ◽  
pp. 013514 ◽  
Author(s):  
M. C. Nucci ◽  
P. G. L. Leach
Keyword(s):  
Jacobi’S Last Multiplier ◽  
Painlevé Vi Equation

Exceptional solutions to the Painlevé VI equation associated with the generalized Jacobi weight

2017 ◽  
Vol 06 (01) ◽  
pp. 1750003
Author(s):  
Shulin Lyu ◽  
Yang Chen
Keyword(s):  
Asymptotic Expansions ◽  
Hankel Determinant ◽  
Generalized Jacobi Weight ◽  
Jacobi Weight ◽  
Special Cases ◽  
Painlevé Vi Equation

We consider the generalized Jacobi weight [Formula: see text], [Formula: see text]. As is shown in [D. Dai and L. Zhang, Painlevé VI and Henkel determinants for the generalized Jocobi weight, J. Phys. A: Math. Theor. 43 (2010), Article ID:055207, 14pp.], the corresponding Hankel determinant is the [Formula: see text]-function of a particular Painlevé VI. We present all the possible asymptotic expansions of the solution of the Painlevé VI equation near [Formula: see text] and [Formula: see text] for generic [Formula: see text]. For four special cases of [Formula: see text] which are related to the dimension of the Hankel determinant, we can find the exceptional solutions of the Painlevé VI equation according to the results of [A. Eremenko, A. Gabrielov and A. Hinkkanen, Exceptional solutions to the Painlevé VI equation, preprint (2016), arXiv:1602.04694 ], and thus give another characterization of the Hankel determinant.

scholarly journals CFT approach to the q-Painlevé VI equation

2017 ◽  
Vol 2 (1) ◽  
Author(s):  
M. Jimbo ◽  
H. Nagoya ◽  
H. Sakai
Keyword(s):  
Painlevé Vi Equation
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