On linear transformations preserving the P\'olya Frequency property

TL;DR

This paper proves that specific linear operators maintain the Pólya frequency property and real-rootedness, addressing several conjectures and open problems in combinatorics.

Contribution

It introduces new results on linear transformations that preserve key properties, resolving longstanding conjectures in combinatorics.

Findings

Certain linear operators preserve Pólya frequency property

Preservation of real-rootedness under these operators

Resolution of conjectures by Bóna, Brenti, and Reiner-Welker

Abstract

We prove that certain linear operators preserve the P\'olya frequency property and real-rootedness, and apply our results to settle some conjectures and open problems in combinatorics proposed by B\'ona, Brenti and Reiner-Welker.