Abstract | In this paper, we use the tools of Gröbner 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 Gröbner basis of quadrics. |