Log-concavity and log-convexity of restricted infinite products

International Journal of Number Theory, pp. 1–25

Abstract

In this paper, we provide a classification of the sign distribution of [Formula: see text], where [Formula: see text] We take the product over [Formula: see text] and denote the complement by [Formula: see text], the set of exceptions. When [Formula: see text] and [Formula: see text] is the set of multiples of [Formula: see text], [Formula: see text] represents the number of [Formula: see text]-regular partitions. More generally, let [Formula: see text] satisfy a certain growth condition. We determine the signs of [Formula: see text] for [Formula: see text] large. The signs mainly depend on which of [Formula: see text] belong to the exception set and the residue class of [Formula: see text] modulo [Formula: see text], where [Formula: see text] depends on [Formula: see text]. For example, let [Formula: see text] and [Formula: see text] is an exception. Let [Formula: see text] be large. Then for almost all [Formula: see text] we have [Formula: see text] If we assume [Formula: see text] and [Formula: see text] is an exception. Let [Formula: see text] be large. Then for almost all [Formula: see text] we have [Formula: see text] Note that this property is independent of the integers [Formula: see text].

Authors 3

1. Krystian Gajdzica

   Jagiellonian University

   Affiliation as printed

   Theoretical Computer Science Department, Faculty of Mathematics and Computer Science, Jagiellonian University, Łojasiewicza 6, 30–348 Krakw, Poland

2. Bernhard E. Heim Aachen Lehrstuhl A für Mathematik

   RWTH Aachen University · University of Cologne

   Affiliation as printed

   Department of Mathematics and Computer Science, Division of Mathematics, University of Cologne, Weyertal 86–90, 50931 Cologne, Germany

   Lehrstuhl A für Mathematik, RWTH Aachen University, 52056 Aachen, Germany

3. Markus Neuhauser Aachen Lehrstuhl für Geometrie und Analysis

   RWTH Aachen University · Kutaisi International University

   Affiliation as printed

   Kutaisi International University, 5/7, Youth Avenue, 4600 Kutaisi, Georgia

   Lehrstuhl für Geometrie und Analysis, RWTH Aachen University, 52056 Aachen, Germany