Inverses of $r$-primitive $k$-normal elements over finite fields

Let $r$, $n$ be positive integers, $k$ be a non-negative integer and $q$ be any prime power such that $r\mid q^n-1.$ An element $\alpha$ of the finite field ${\mathbb{F}}_{q^n}$ is called an {\it $r$-primitive} element, if its multiplicative order is $(q^n-1)\mathrm{/}r$, and it is called a {\it $k$-normal} element over ${\mathbb{F}}_q$, if the greatest common divisor of the polynomials $m_{\alpha }(x)={\sum }_{i=1}^n{\alpha }^{q^{i-1}}{x}^{n-i}$ and $x^n-1$ is of degree $k\mathrm{.}$ In this article, we define the characteristic function for the set of $k$-normal elements, and with the help of this, we establish a sufficient condition for the existence of an element $\alpha$ in ${\mathbb{F}}_{q^n}$, such that $\alpha$ and ${\alpha }^{-1}$ both are simultaneously $r$-primitive and $k$-normal over ${\mathbb{F}}_q$. Moreover, for $n>6k$, we show that there always exists an $r$-primitive and $k$-normal element $\alpha$ such that ${\alpha }^{-1}$ is also $r$-primitive and $k$-normal in all but finitely many fields ${\mathbb{F}}_{q^n}$ over ${\mathbb{F}}_q$, where $q$ and $n$ are such that $r\mid q^n-1$andthereexistsa$k$-degreepolynomial$g(x)\mid x^n-1$over$$\mathbb{F}$q$\mathrm{.}Inparticular,wediscusstheexistenceofanelement$\alpha$in$$\mathbb{F}${q^n}$suchthat$\alpha$and$\alpha^{-1}$ both are simultaneously $1$-primitive and $1$-normal over ${\mathbb{F}}_q$.