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An ultradiscrete analog of the Bessel function is constructed by taking theultradiscrete limit for a $q$-difference analog of the Bessel function. Then, adirect relationship between a class of special solutions for the ultradiscretePainlev\'{e} III equation and those of the discrete Painlev\'{e} III equationwhich have a determinantal structure is established.

Ultradiscrete limit of Bessel function type solutions of the Painlev\'{e} III equation