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Painlevé versus Fuchs. (English) Zbl 1119.34069
Summary: The sigma form of the Painlevé VI (PVI) equation contains four arbitrary parameters and generically the solutions can be said to be genuinely ’nonlinear’ because they do not satisfy linear differential equations of finite order. However, when there are certain restrictions on the four parameters, there exist one-parameter families of solutions which do satisfy (Fuchsian) differential equations of finite order. We study this phenomenon of Fuchsian solutions to the Painlevé equation with a focus on the particular PVI equation which is satisfied by the diagonal correlation function $C\left(N,N\right)$ of the Ising model. We obtain Fuchsian equations of order $N+1$ for $C\left(N,N\right)$ and show that the equation for $C\left(N,N\right)$ is equivalent to the Nth symmetric power of the equation for the elliptic integral $E$. We show that these Fuchsian equations correspond to rational algebraic curves with an additional Riccati structure and we show that the Malmquist Hamiltonian $p,q$ variables are rational functions in complete elliptic integrals. Fuchsian equations for off-diagonal correlations $C\left(N,M\right)$ are given which extend our considerations to discrete generalizations of Painlevé.
##### MSC:
 34M55 Painlevé and other special equations; classification, hierarchies 33E17 Painlevé-type functions 47E05 Ordinary differential operators