Polynomials interpolated totally positive sequences

A real sequence $(a_k)_{k=0}^\infty$ is called {\it totally positive} if all minors of the infinite Toeplitz matrix $\left\| a_{j-i} \right\|_{i, j =0}^\infty$ are nonnegative (where $a_k=0$ for $k<0$). In this paper, we investigate the following question: for which real polynomials $P$ the sequence $(P(k))_{k=0}^\infty$ is totally positive? We establish a few new necessary conditions, sufficient conditions, present a number of important examples and formulate several open problems.