An analog of multiplier sequences for the set of totally positive sequences

Abstract

A real sequence $(b_k{)}_{k=0}^{\infty }$ is called totally positive if all minors of the infinite matrix ${\Vert b_{j-i}\Vert }_{i,j=0}^{\infty }$ are nonnegative (here $b_k=0$ for $k<0$). In this paper, we investigate the problem of description of the set of sequences $(a_k{)}_{k=0}^{\infty }$ such that for every totally positive sequence $(b_k{)}_{k=0}^{\infty }$ the sequence $(a_k{b}_k{)}_{k=0}^{\infty }$ is also totally positive. We obtain the description of such sequences $(a_k{)}_{k=0}^{\infty }$ in two cases: when the generating function of the sequence ${\sum }_{k=0}^{\infty }{a}_k{z}^k$ has at least one pole, and when the sequence $(a_k{)}_{k=0}^{\infty }$ has not more than $4$ nonzero terms.