On the Log-Concavity of the D'Arcais Polynomials for Normalised Functions

Johann Stumpenhusen

number theory

On the Log-Concavity of the D'Arcais Polynomials for Normalised Functions

arXiv:2607.14961

summary

The paper introduces a new variant of log‑concavity for families of polynomials and proves that specific D'Arcais polynomials satisfy this property at particular points.

Abstract

A sequence $(a_{n})_{n \in N}$ of non-negative real numbers is called log-concave at $n$ if $a_{n}^{2} \geq a_{n + 1} a_{n - 1}$ . This property has been generalised in various ways to families of polynomials. We introduce a new variant and show that certain types of D'Arcais polynomials have the respective properties at certain points.

8 pages

Topics & keywords

#log-concavity #d'arcais polynomials #polynomial sequences #combinatorial number theory #generating functionslog-concaveD'Arcais polynomialsnormalized functionssequence inequalitiespolynomial families