Elements of Polya-Schur theory in finite difference setting

2012/04/13 by Petter Brändén, P. Brändén, Ilia Krasikov +6 · 2 citations
Mathematics · #Advanced Combinatorial Mathematics #Advanced Topics in Algebra #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical functions and polynomials #math.CA

paper · pdf · doi:10.48550/arxiv.1204.2963

11 pages, 2 figures (Substantial revision of the previous version, material on generalized Laguerre inequalities removed, results on discrete multiplier sequences strengthened)

openalex publication_date 2012/04/13 · arxiv created 2013/06/23 · arxiv updated 2013/06/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this note we attempt to develop an analog of Pólya-Schur theory describing the class of univariate hyperbolicity preservers in the setting of linear finite difference operators. We study the class of linear finite difference operators preserving the set of real-rooted polynomials whose mesh (i.e. the minimal distance between the roots) is at least one. In particular, finite difference versions of the classical Hermite-Poulain theorem and generalized Laguerre inequalities are obtained.

Citations

• Discrete analogues of the Laguerre inequalities and a conjecture of I. Krasikov
• Extended Laguerre inequalities and a criterion for real zeros

Cited by

• Linear finite difference operators with constant coefficients and distribution of zeros of polynomials
• Linear finite difference operators preserving Laguerre-Pólya class
• Duality, polarity and convolution in umbral calculus

Related

• Linear finite difference operators preserving Laguerre-Pólya class
• Preserving positive polynomials and beyond
• Hermite-Poulain theorems for linear finite difference operators
• An inverse problem in Polya-Schur theory. I. Non-genegerate and degenerate operators
• Constructive solutions to Pólya-Schur problems