Lacunary partition functions
type: Item
instance of: scholarly article
title: Lacunary partition functions
zbMATH Open document ID: 1007.11060
author: Jeremy Lovejoy
published in: Mathematical Research Letters
publication date: 2002-12-01T00:00:00Z
review text: A function \(f(q)\) is lacunary if \(f(q)= \sum_{n\geq 0} a(n)q^n\) and \(a(n)\) is almost always \(0\). It is known by the work of J.-P. Serre, B. Gordon and S. Robins that there are approximately 60 pairs \((r,s)\) for which \[ \prod_{n=1}^\infty (1-q^n)^r(1-q^{2n})^s \] is lacunary. In this article, the author first establishes, using the theory of Bailey pairs, identities expressing multi-sum Rogers-Ramanujan type expressions in terms of Hecke type functions. An example is \[ \begin{split} &\sum_{n\geq 0}\sum_{j=-n}^n(-1)^{n+j}q^{((2k+3)n^2+(2k+1)n)/2-j(3j+1)/2}(1-q^{2n+1}) \\ &\qquad = \sum_{n_k\geq n_{k-1}\geq\cdots \geq n_1\geq 0} \frac{(q;q)_{n_k}(-1)^{n_k}q^{n_k(n_k+1)/2+n_{k-1}(n_{k-1}+1)+\cdots +n_1(n_1+1)}}{(q;q)_{n_k-n_{k-1}}\cdots (q;q)_{n_2-n_1}},\end{split}\tag{1} \] where as usual \[ (a;q)_n = (1-a)(1-aq)\cdots (1-aq^{n-1}). \] Using a result due to P. Bernays, the author deduces that the left hand side of (1) is lacunary. He then gives a partition-theoretic interpretation of the right hand side of (1) and explains why the difference of certain partition functions is almost always 0. Such results are analogous to Euler's result, which states that if \(E(n)\) and \(O(n)\) denote respectively the number of partitions of \(n\) into an even and odd number of distinct parts, then \(E(n)-O(n)=0\) for almost all \(n\).
reviewed by: Heng Huat Chan
Mathematics Subject Classification ID: 11P82
Mathematics Subject Classification ID: 05A17
Mathematics Subject Classification ID: 33D15
Mathematics Subject Classification ID: 30B10
zbMATH DE Number: 1804047
zbMATH Keywords: Bailey pairs
zbMATH Keywords: lacunary series
MaRDI profile type: Publication
full work available at URL: mrl.2002.v9.n2.a5
OpenAlex ID: W2030654046
DOI: 10.4310/MRL.2002.V9.N2.A5
description: scientific article; zbMATH DE number 1804047