Polynomials interpolated totally positive sequences

Abstract

A real sequence $(a_k{)}_{k=0}^{\infty }$ is called {\it totally positive} if all minors of the infinite Toeplitz matrix ${\Vert a_{j-i}\Vert }_{i,j=0}^{\infty }$ 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.