The DP color function \(P_{DP}({\mathcal {H}},k)\) of a hypergraph \({\mathcal {H}}\) for \(k\in \mathbb N\) is introduced, based on the DP coloring introduced by Bernshteyn and Kostochka, which is the minimum value of \(P_{DP}(\mathcal {H},\mathcal {F})\) over all k-fold covers \(\mathcal {F}\) of \(\mathcal {H}\) . It is an extension of the chromatic polynomial of \({\mathcal {H}}\) with the property that \(P_{DP}({\mathcal {H}},k)\le P({\mathcal {H}},k)\) for all positive integers k. In this article, we obtain an upper bound for \(P_{DP}({\mathcal {H}},k)\) when \({\mathcal {H}}\) is a connected r-uniform hypergraph for \(r\ge 2\) , and the upper bound is attained if and only if \({\mathcal {H}}\) is an r-uniform hypertree. We also show that \(P_{DP}({\mathcal {H}},k)= P({\mathcal {H}},k)\) holds when \({\mathcal {H}}\) is an r-uniform hypertree or a unicyclic linear r-uniform hypergraph with an odd cycle for \(r\ge 3\) . These conclusions coincide with the known results of graphs.
