Bent functions, which are maximally nonlinear Boolean functions with even numbers of variables and whose Hamming distance to the set of all affine functions equals $2^{n-1}pm 2^{frac{n}{2}-1}$, were introduced by Rothaus in 1976 when he considered problems in combinatorics. Bent functions have been extensively studied due to their applications in cryptography, such as S-box, block cipher and stream cipher. Further, they have been applied to coding theory, spread spectrum and combinatorial design. Hyper-bent functions, as a special class of bent functions, were introduced by Youssef and Gong in 2001, which have stronger properties and rarer elements. Many research focus on the construction of bent and hyper-bent functions. In this paper, we consider functions defined over $mathbb{F}_{2^n}$ by $f_{a,b}:=mathrm{Tr}_{1}^{n}(ax^{(2^m-1)})+mathrm{Tr}_{1}^{4}(bx^{frac{2^n-1}{5}})$, where $n=2m$, $mequiv 2pmod 4$, $ain mathbb{F}_{2^m}$ and $binmathbb{F}_{16}$. When $ain mathbb{F}_{2^m}$ and $(b+1)(b^4+b+1)=0$, with the help of Kloosterman sums and the factorization of $x^5+x+a^{-1}$, we present a characterization of hyper-bentness of $f_{a,b}$. Further, we use generalized Ramanujan-Nagell equations to characterize hyper-bent functions of $f_{a,b}$ in the case $ainmathbb{F}_{2^{frac{m}{2}}}$.
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