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We study a three-point mapping that has the Painlevé-II equation as continuous limit. The qualifier `alternate' is due to the fact that there already exists another discrete form of . We present the Lax pair for this discrete equation, as well as its auto-Bäcklund transform. Special rational solutions as well as solutions in terms of the (discrete) Airy function are obtained for particular values of the parameter of the equation.
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Nijhoff et al. (1996) studied this question.
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