Ramanujan-type partial theta identities and conjugate Bailey pairs
type: Item
instance of: scholarly article
title: Ramanujan-type partial theta identities and conjugate Bailey pairs
zbMATH Open document ID: 1322.11107
author: Jeremy Lovejoy
published in: The Ramanujan Journal
publication date: 2013-01-02T00:00:00Z
review text: In his lost notebook, Ramanujan recorded a number of partial theta identities such as those appearing in [\textit{G. E. Andrews} and \textit{B. C. Berndt}, Ramanujan's lost notebook. Part II. New York, NY: Springer (2009; Zbl 1180.11001), Entry 6.6.1]; among them, we have \[ \sum_{n \geq 0} \dfrac{(q^{n+1})_{n} q^{n}} {(-a)_{n+1}(-q/a)_{n}} = \sum_{n \geq 0} (-a)^{n} q^{n^{2}+n} - \dfrac{a}{(-a,-q/a)_{\infty}} \sum_{n \geq 0} (-1)^{n} a^{3n} q^{3n^2+2n}(1+aq^{2n+1}). \] Here we employ the usual basic hypergeometric series notation, namely \[ (a)_{n} := (a;q)_{n} = (1-a)(1-aq) \cdots (1-aq^{n-1}), \quad (a_{1},a_{2},\ldots, a_{j})_{n} : = (a_{1};q)_{n}(a_{2};q)_{n} \cdots (a_{j};q)_{n}, \] and \[ (a)_{\infty} = \lim_{n \to \infty} (a)_{n} \] when \(|q| < 1\). Ramanujan's partial theta identities were first proven by \textit{G. E. Andrews} [Adv. Math. 41, 137--172 (1981; Zbl 0477.33001)]. Subsequently, \textit{S. O. Warnaar} [Proc. Lond. Math. Soc. (3) 87, No. 2, 363--395 (2003; Zbl 1089.05009)] placed the identities in the context of Bailey pairs and demonstrated how to generate further examples. A Bailey pair relative to \((a,\, q)\) is a pair of sequences \((\alpha_{n},\, \beta_{n})_{n \geq 0}\) satisfying \[ \beta_{n} = \sum_{r=0}^{n} \dfrac{\alpha_{r}}{(aq)_{n+r}(q)_{n-r}}. \] A conjugate Bailey pair relative to \((a,\, q)\) is a pair of sequences \((\delta_{n},\, \gamma_{n})_{n \geq 0}\) satisfying \[ \gamma_{n} = \sum_{r=n}^{\infty} \dfrac{\delta_{r}}{(aq)_{r+n}(q)_{r-n}}. \] The author demonstrates how Ramanujan-type partial theta identities give rise, through their residual identities, to conjugate Bailey pairs. To illustrate his ideas, the author proves six partial theta identities of Ramanujan and five identities stated by Warnaar; the author additionally derives six further identities using Warnaar's method. It is known [\textit{W. N. Bailey}, Proc. Lond. Math. Soc. (2) 50, 1--10 (1948; Zbl 0031.39203)] that a Bailey pair and a conjugate Bailey pair satisf…
reviewed by: Chadwick Gugg
Mathematics Subject Classification ID: 11P83
Mathematics Subject Classification ID: 33D15
Mathematics Subject Classification ID: 11F27
zbMATH DE Number: 6120461
zbMATH Keywords: Bailey pairs
zbMATH Keywords: conjugate Bailey pairs
zbMATH Keywords: partial theta identities
zbMATH Keywords: mixed mock modular forms
MaRDI profile type: Publication
full work available at URL: s11139-011-9356-4
OpenAlex ID: W1967309294
cites work: Ramanujan's ``lost'' notebook. I: Partial Theta-functions
cites work: The Fifth and Seventh Order Mock Theta Functions
cites work: Q3724706
cites work: Bailey chains and generalized Lambert series. I: Four identities of Ramanujan
cites work: Partitions and indefinite quadratic forms.
cites work: Ramanujan's ``Lost'' Notebook. VII: The sixth order mock theta functions
cites work: The Bailey transform and false theta functions
cites work: Identities of the Rogers-Ramanujan Type
cites work: Ramanujan's Lost Notebook
cites work: Some identities for terminating q-series
cites work: Harmonic Maaß-Jacobi forms of degree 1 with higher rank indices
cites work: Q4040189
cites work: Q4828543
cites work: Hecke-type double sums, Appell-Lerch sums, and mock theta functions. I.
cites work: Hecke Type Formula for Unified Witten–Reshetikhin–Turaev Invariants as Higher-Order Mock Theta Functions
cites work: Lacunary partition functions
cites work: A Bailey lattice
cites work: A new general conjugate Bailey pair
cites work: Q3152500
cites work: A NOTE ON A TRANSFORMATION OF BAILEY
cites work: A New Proof of Rogers's Transformations of Infinite Series
cites work: Partial Theta Functions. I. Beyond the Lost Notebook
cites work: Q3566608
cites work: Q3590677
DOI: 10.1007/S11139-011-9356-4
description: scientific article; zbMATH DE number 6120461