Log-concavity of infinite product generating functions

Abstract

In the 1970s Nicolas proved that the coefficients p_d(n) defined by the generating function ∑n=0∞pd(n)qn=∏n=1∞1-qn-nd-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\sum _{n=0}^{\\infty } p_d(n) \\, q^n = \\prod _{n=1}^{\\infty } \\left( 1- q^n\\right) ^{-n^{d-1}} \\end{aligned}$$\\end{document}are log-concave for d=1. Recently, Ono, Pujahari, and Rolen have extended the result to d=2. Note that p_1(n)=p(n) is the partition function and p_2(n)=mathrm{pp}left( nright) is the number of plane partitions. In this paper, we invest in properties for p_d(n) for general d. Let n ge 6. Then p_d(n) is almost log-concave for n divisible by 3 and almost strictly log-convex otherwise.