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An element α ∈ F q n is a normal element over F q if the conjugates α q i , 0 ≤ i ≤ n − 1, are linearly independent over F q . Hence a normal basis for F q n over F q is of the form {α, α q , . . . , α q n−1 }, where α ∈ F q n is normal over F q .In 2013, Huczynska, Mullen, Panario and Thomson introduce the concept of knormal elements, as a generalization of the notion of normal elements. In the last few years, several results have been known about these numbers. In this paper, we give an explicit combinatorial formula for the number of k-normal elements in the general case, answering an open problem proposed by Huczynska et al. (2013).

Let F q n be a finite field with q n elements and r be a positive divisor of q n − 1. An element α ∈ F * q n is called r-primitive if its multiplicative order is (q n −1)/r. Also, α ∈ F q n is k-normal over F q if the greatest common divisor of the polynomials g α (x) = αx n−1 +α q x n−2 +. . .+α q n−2 x+α q n−1 and x n −1 in F q n [x] has degree k. These concepts generalize the ideas of primitive and normal elements, respectively. In this paper, we consider non-negative integers m 1 , m 2 , k 1 , k 2 , positive integers r 1 , r 2 and rational functions F (x) = F 1 (x)/F 2 (x) ∈ F q n (x) with deg(F i ) ≤ m i for i ∈ {1, 2} satisfying certain conditions and we present sufficient conditions for the existence of r 1 -primitive k 1 -normal elements α ∈ F q n over F q , such that F (α) is an r 2 -primitive k 2 -normal element over F q . Finally as an example we study the case where r 1 = 2, r 2 = 3, k 1 = 2, k 2 = 1, m 1 = 2 and m 2 = 1, with n ≥ 7.