A recursion formula for the generalized Euler function $φ_e(n)$

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 $φ_e(n)$, which is determined by some matrices related to a congruence equation modulo $e$. Furthermore, through the recursion formula, we get the explicit formula for $φ_5(n)$. Our results generalize the corresponding results in \cite{A4,A8,A10,A11}.

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