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

Olga Katkova, Anna Vishnyakova

Abstract

A real sequence $(b_k)_{k=0}^\infty$ is called totally positive if all minors of the infinite matrix $ \left\| b_{j-i} \right\|_{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.

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

Abstract

A real sequence is called totally positive if all minors of the infinite matrix are nonnegative (here for ). In this paper, we investigate the problem of description of the set of sequences such that for every totally positive sequence the sequence is also totally positive. We obtain the description of such sequences in two cases: when the generating function of the sequence has at least one pole, and when the sequence has not more than nonzero terms.
Paper Structure (3 sections, 5 theorems, 93 equations)

This paper contains 3 sections, 5 theorems, 93 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['tt1']}
  3. Proof of Theorem \ref{['tt2']}

Key Result

Theorem 1.7

Let ${\mathbf A} =(a_k)_{k=0}^\infty$ be a nonnegative sequence, and its generating function is a meromorphic function with at least one pole. Then for every $(b_k)_{k=0}^\infty \in TP_\infty$ we have $\Lambda_{\mathbf A} ((b_k)_{k=0}^\infty) \in TP_\infty$ if and only if $A(z) = \frac{C}{1- \beta

Theorems & Definitions (13)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Definition 1.5
  • Theorem 1.7
  • Theorem 1.9
  • Theorem 1.10
  • Example 1.11
  • Conjecture 1.12
  • ...and 3 more