Linearization And Solutions Of The Discrete Painlevé-III Equation

Abstract

We present particular solutions of the discrete Painlevé III (d-P${}_{\mathrm{III}}$) equation of rational and special function (Bessel) type. These solutions allow us to establish a close parallel between this discrete equation and its continuous counterpart. Moreover, we propose an alternate form for d-P${}_{\mathrm{III}}$ and confirm its integrability by explicitly deriving its Lax pair.

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