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

Abstract

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}.