A recursion formula for the generalized Euler function ${\phi }_e(n)$

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 ${\phi }_e(n)$, which is determined by some matrices related to a congruence equation modulo $e$. Furthermore, through the recursion formula, we get the explicit formula for ${\phi }_5(n)$. Our results generalize the corresponding results in \cite{A4,A8,A10,A11}.