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