Coalescence limits for higher order Painlevé equations

It is well-known that the first Painleve equation arises as a coalescence limit of each of the other five Painleve equations. This result is important because it shows that, since the solution of the first Painleve equation cannot be expressed in terms of known functions, then neither can the solutions of the other five Painleve equations (except possibly for special values of their parameters). Here we derive analogous results for three recently derived higher order ordinary differential equations believed to define new transcendental functions. We show that each of the equations considered has as a coalescence limit a member of the first Painleve hierarchy. We thus reduce the problem of showing that the solutions of these three cannot be expressed in terms of known functions to that of showing that the same is true for the corresponding first Painleve equations. This represents the first extension of coalescence results for the Painleve equations to their higher order analogues.

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