Horizontal and Vertical Log-Concavity

TL;DR

This paper explores the properties of log-concavity in various triangular sequences derived from important polynomials across combinatorics, number theory, and physics, extending previous work on generating functions and recursion relations.

Contribution

It introduces a comprehensive study of horizontal and vertical log-concavity for a broad class of polynomial-based triangular sequences, including Laguerre, Pochhammer, D'Arcais, and Nekrasov-Okounkov polynomials.

Findings

Identifies log-concavity properties in several polynomial sequences

Provides new insights into the structure of these sequences in combinatorics and physics

Extends previous work on generating functions and recursion relations

Abstract

Horizontal and vertical generating functions and recursion relations have been investigated by Comtet for triangular double sequences. In this paper we investigate the horizontal and vertical log-concavity of triangular sequences assigned to polynomials which show up in combinatorics, number theory and physics. This includes Laguerre polynomials, the Pochhammer polynomials, the D'Arcais and Nekrasov--Okounkov polynomials.