Derivation module and the Hilbert-Kunz multiplicity of the co-ordinate ring of a projective monomial curve

Let $n_0,n_1,\dots ,n_p$ be a sequence of positive integers such that $n_0<n_1<\cdots <n_p$ and $\mathrm{g}\mathrm{c}\mathrm{d}(n_0,n_1,\dots ,n_p)=1$. Let $S=⟨(0,n_p),(n_0,n_p-n_0),\dots ,(n_{p-1},n_p-n_{p-1}),(n_p,0)⟩$ be an affine semigroup in ${\mathbb{N}}^2$. The semigroup ring $k[S]$ is the co-ordinate ring of the projective monomial curve in the projective space ${\mathbb{P}}_k^{p+1}$, which is defined parametrically by \begin{center} $x_0=v^{n_p},x_1=u^{n_0}{v}^{n_p-n_0},\dots ,x_p=u^{n_{p-1}}{v}^{n_p-n_{p-1}},x_{p+1}=u^{n_p}$. \end{center} In this article, we consider the case when $n_0,n_1,\dots ,n_p$ forms an arithmetic sequence, and give an explicit set of minimal generators for the derivation module ${\mathrm{D}\mathrm{e}\mathrm{r}}_k(k[S])$. Further, we give an explicit formula for the Hilbert-Kunz multiplicity of the co-ordinate ring of a projective monomial curve.