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

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