Generation of Painlevé V transcendents

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Abstract

An algorithm for generating solutions to the Painlevé V equation (the Painlevé V transcendents) is presented. The first step is to look for general one-dimensional Schrödinger Hamiltonians ruled by third degree polynomial Heisenberg algebras, which have fourth order differential ladder operators. It is realized then that there is a key function that must satisfy the Painlevé V equation. Conversely, by identifying systems ruled by a third degree polynomial Heisenberg algebra, in particular their four extremal states, this key function can be built straightforwardly. The simplest Painlevé V transcendents will be generated through this algorithm.

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Bermudez, D., Fernández, D. J., & Negro, J. (2019). Generation of Painlevé V transcendents. In Trends in Mathematics (pp. 24–33). Springer. https://doi.org/10.1007/978-3-030-34072-8_3

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