The polar transformation of a binary erasure channel (BEC) can be exactlyapproximated by other BECs. Ar{\i}kan proposed that polar codes for a BEC canbe efficiently constructed by using its useful property. This study proposes anew class of arbitrary input generalized erasure channels, which can be exactlyapproximated the polar transformation by other same channel models, as with theBEC. One of the main results is the recursive formulas of the polartransformation of the proposed channel. In the study, we evaluate the polartransformation by using the $\alpha$-mutual information. Particularly, when theinput alphabet size is a prime power, we examines the following: (i)inequalities for the average of the $\alpha$-mutual information of the proposedchannel after the one-step polar transformation, and (ii) the exact proportionof polarizations of the $\alpha$-mutual information of proposed channels ininfinite number of polar transformations.
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