  In this paper, we use the tools of Gr\"{o}bner bases and combinatorial secant
varieties to study the determinantal ideals $I_t$ of the extended Hankel
matrices. Denote by $c$-chain a sequence $a_1,\...,a_k$ with $a_i+c<a_{i+1}$
for all $i=1,\...,k-1$. Using the results of $c$-chain, we solve the membership
problem for the symbolic powers $I_t^{(s)}$ and we compute the primary
decomposition of the product $I_{t_1}\... I_{t_k}$ of the determinantal ideals.
Passing through the initial ideals and algebras we prove that the product
$I_{t_1}\... I_{t_k}$ has a linear resolution and the multi-homogeneous Rees
algebra $\Rees(I_{t_1},\...,I_{t_k})$ is defined by a Gr\"obner basis of
quadrics.
