  In this paper, basing on the linear algebra methods and elementary
techniques, for any positive integers $ e $ and $ n $, we obtain a recursion
formula for the generalized Euler function $ \varphi_e(n) $, which is
determined by some matrices related to a congruence equation modulo $ e $.
Furthermore, through the recursion formula, we get the explicit formula for $
\varphi_5(n) $. Our results generalize the corresponding results in
\cite{A4,A8,A10,A11}.
