A coalescent diagram of the Painlevé equations from the viewpoint of isomonodromic deformations

Moduli spaces for linear differential equations and the Painlevé equations

“…Of the nine families with irregular singularities, six are again classical [16,4]. The three remaining ones were also recently discovered in [21,20]. The corresponding Painlevé equations appear already in [27] from the viewpoint of the Okamoto-Painlevé pairs.…”

Geometric aspects of Painlevé equations

“…We also use q given by 94) as the multiplicative parameter for the null root δ. The Weyl group actions for the τ variables are the same as (5.31), where the s 3 action simplifies to…”

Meromorphic solutions of P_4,34 and their value distribution

“…As we have already mentioned, in case (C2), putting α = −β/2, β = b 3 , γ = 2d 4 in (7) we obtain that 2df (t), where t = (z + b 3 /(4d 4 ))d is a solution of P 4 (α,β) with parametersα = b 6 d −6 /16 = β 2 /(4 √ 2γ 3/2 ) andβ = β [20]. Clearly, by this change of variables one can find the Bäcklund transformations, one-parameter and rational solutions for special values of the parameters of P 4,34 (−β/2, β, γ) coming from the corresponding transformations and solutions of P 4 [8].…”

“…Equation P 4,34 admits the following scaling transformation: if f (z) is a solution of P 4,34 (α, β, γ), then f (cz)/c is a solution of P 4,34 (α, c 3 β, c 4 γ) [20]. It is shown in [20] that if β = 0, γ = 0, then equation (7) can easily be integrated with polynomial solutions…”

“…If γ = 0, β = 0, then, by changing the independent variable t → −t in (7) one obtains (6) with parameters A = α and B = β. 20]. The properties of P 4,34 and P 34 not connected directly to the Nevanlinna theory will be discussed in the Appendix.…”