Log-Concavity of Infinite Product Generating Functions

2022/02/01 by Bernhard Heim, Markus Neuhäuser, Heim, Bernhard +1 · 1 citation
Mathematics · #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Analytic Number Theory Research #Combinatorics (math.CO) #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.2202.00627

openalex publication_date 2022/02/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In the 1970s Nicolas proved that the coefficients p_d(n) defined by the generating function ∑_n=0^∞ p_d(n) qⁿ = ∏_n=1^∞ ( 1- qⁿ)^-n^d-1 are log-concave for d=1. Recently, Ono, Pujahari, and Rolen have extended the result to d=2. Note that p₁(n)=p(n) is the partition function and p₂(n)=\funcpp( n) is the number of plane partitions. In this paper, we invest in properties for p_d(n) for general d. Let n ≥ 6. Then p_d(n) is almost log-concave for n divisible by 3 and almost strictly log-convex otherwise.

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• Log-Concavity and Log-Convexity of Restricted Infinite Products

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