  The generalized group of units of the ring modulo $n$ was first introduced by
El-Kassar and Chehade, written as $U^k(Z_n)$. This allows us to formulate a new
generalization to the Euler phi function $\varphi(n)$, that represents the
order of $% U^k(Z_n)$ and it is denoted by $\varphi ^{k}(n).$ In this paper, we
introduce this newly defined function, where we compute its explicit form and
examine some of its properties similar to that of $\varphi (n)$. In addition,
we study some generalized equations involving $\varphi ^{k}(n)$ where complete
solution is given for some equations by considering the general case and others
for some particular cases.
