A note on the $ c $-differential spectrum of an AP$ c $N function

Motivated by a recent work of Zhang and Yan on the \begin{document}$ c $\end{document}-differential spectrum some power functions over finite fields, we further study an AP\begin{document}$ $\end{document}N function express its id="M5">\begin{document}$ in terms id="M6">\begin{document}$ (i, j, k)_2 $\end{document}, i.e., cardinality intersection id="M7">\begin{document}$ (\mathcal{C}^{(2)}_i+1)\cap\mathcal{C}^{(2)}_j\cap(\mathcal{C}^{(2)}_k-1) $\end{document} for id="M8">\begin{document}$ i, k\in\{0, 1\} where id="M9">\begin{document}$ \mathcal{C}^{(2)}_0, \mathcal{C}^{(2)}_1 are cyclotomic classes order two field id="M10">\begin{document}$ \mathbb{F}_{p^n} id="M11">\begin{document}$ p is odd prime id="M12">\begin{document}$ n positive integer. By virtue numbers orders four, determine values id="M13">\begin{document}$ id="M14">\begin{document}$ which may be independent interest. As application, give another proof id="M15">\begin{document}$ id="M16">\begin{document}$ fields with characteristic id="M17">\begin{document}$ 5 $\end{document}. Further, refine result sense that completely characterize conditions when id="M18">\begin{document}$ equation id="M19">\begin{document}$ has one solution solutions, respectively.