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We study the commuting problem for Toeplitz operators on the Hardy space of the polydisk. We consider two symbols where one is an arbitrarily bounded symbol and the other is a bounded symbol which has the form of a sum of an analytic function and a co-analytic function. We then give a necessary and sufficient condition on the symbols for which the corresponding Toeplitz operators commute each other.

Commuting Toeplitz operators on the Hardy space of the polydisk