Linear resolutions and Gr\"{o}bner basis of Hankel determinantal ideals

In this paper, we study the family of determinantal ideals of "close" cuts of Hankel matrices, say $\mathcal{f}$. We show that the multi-Rees algebra of ideals in $\mathcal{f}$ is defined by a quadratic Gr\"{o}bner basis, it is Koszul, normal Cohen-Macaulay domain and it has a nice Sagbi basis. As a consequence of Koszulness, we prove that every product $I_1\ldots I_l$ of ideals of $\mathcal{f}$ has linear resolution. Moreover, we show that natural generators of every product $I_1\ldots I_l$ form a Gr\"{o}bner basis.