A note on the rank parity function
type: Item
instance of: scholarly article
title: A note on the rank parity function
zbMATH Open document ID: 1228.11158
author: Alexander E. Patkowski
published in: Discrete Mathematics
publication date: 2010-04-27T00:00:00Z
review text: The rank of a partition is defined as the largest part minus the number of parts. Let \(S(n)\) denote the number of partitions into distinct parts with even rank minus those with odd rank. In the paper under review the author proves the following identity for positive even integers \(n\): \[ S\left(\frac{n}{2}\right)=\sum_{\substack{ r\geq 0,\, |j|\leq r, \\ 24n+2=3(4r+1)^2-(6j+1)^2 }} (-1)^{r+j}-\sum_{\substack{ r\geq 0,\, |j|\leq r, \\ 24n+2=3(4r+3)^2-(6j+1)^2 }} (-1)^{r+j}. \] The author also establishes some results for the q-series related with the generating functions of \(S_E(n)\) and \(S_O(n)\), where \(S_E(n)\) (resp. \(S_O(n)\)) is the number of partitions of \(n\) into distinct parts with even rank minus the number of them with odd rank where the number of parts is even (resp. odd). The proofs of these results are achieved using two new Bailey pairs.
reviewed by: Ricardo Mamede
Mathematics Subject Classification ID: 11P81
Mathematics Subject Classification ID: 05A15
Mathematics Subject Classification ID: 05A17
Mathematics Subject Classification ID: 11F27
zbMATH DE Number: 5701955
zbMATH Keywords: partitions
zbMATH Keywords: q-series
zbMATH Keywords: indefinite quadratic forms
MaRDI profile type: Publication
full work available at URL: j.disc.2009.10.001
OpenAlex ID: W2090816779
cites work: Enumerative proofs of certain q-identities
cites work: The Fifth and Seventh Order Mock Theta Functions
cites work: ON q-DIFFERENCE EQUATIONS FOR CERTAIN WELL-POISED BASIC HYPERGEOMETRIC SERIES
cites work: Partitions and indefinite quadratic forms.
cites work: Q3023367
cites work: Partitions With Distinct Evens
cites work: Identities of the Rogers-Ramanujan Type
cites work: Change of base in Bailey pairs
cites work: Multiplicative $q$-hypergeometric series arising from real quadratic fields
cites work: Characters and \(q\)-series in \(\mathbb Q(\sqrt 2)\)
cites work: Q4040189
cites work: Q4040884
cites work: Lacunary partition functions
cites work: On Some Partition Functions Related to Some Mock Theta Functions
cites work: Extensions of the well-poised and elliptic well-poised Bailey lemma
DOI: 10.1016/J.DISC.2009.10.001
description: scientific article; zbMATH DE number 5701955