Elementary proofs of some \(q\)-identities of Jackson and Andrews--Jain
type: Item
instance of: scholarly article
title: Elementary proofs of some \(q\)-identities of Jackson and Andrews--Jain
zbMATH Open document ID: 1080.33015
author: Victor J. W. Guo
published in: Discrete Mathematics
publication date: 2005-06-23T00:00:00Z
review text: The author presents proofs of three \(q\)-identities of Jackson. They are Jackson's terminating \(q\)-analogue of Dixon's sum, Jackson's \(q\)-analogue of Clausen's formula, and a generalization of both of them. He then proceeds to proofs of Andrews's so-called \(q\)-analogue of a terminating version of Watson's summation formula for \(\;_{3}F_{2}\), formula 1.4 in Guo's paper, which is claimed to be equivalent to Jain's formula 1.5. The correct version of 1.4. is found in \textit{G. E. Andrews}' paper [(*) ``On \(q\)-analogues of the Watson and Whipple summations'', SIAM J. Math. Anal. 7, 332--336 (1976; Zbl 0339.33007),]. Now there are two possibilities. We start with Andrews' correct version of 1.4 (see (*)) and let \(c\to{-\infty}\). We then arrive at Andrews' \(q\)-analogue of \textit{E. E. Kummer}'s formula [J. Reine Angew. Math. 15, 39--83 and 127--172 (1836; ERAM 015.0533cj)] from \textit{G. E. Andrews} [Duke Math. J. 40, 525--528 (1973; Zbl 0266.33003)]. We start with the Guo's formula 1.4., which can also be found in (*) just below the correct version. Now let \(c\to{-\infty}\). We then arrive at the other version of the \(q\)-analogue of Kummer's formula from \textit{A. Verma} and \textit{V. K. Jain} [Ganita 43, No. 1--2, 1--31 (1992; Zbl 0835.33009)]. Formula 1.5 is not equivalent to 1.4 because the meaning of integer variable is changed between the equations. The author also mentions his symmetric \(q\)-Pfaff-Saalschütz identity from \textit{V. J. W. Guo} and \textit{J. A. Zeng} [Electron. J. Comb. 12, No. 1, Research paper N2 (2005; Zbl 1060.05011)]. However the reviewer couldn't see any symmetric \(q\)-Pfaff-Saalschütz identity in that paper. Reviewer's remark: In summary: the current status of \(q\)-calculus is that of multiplication before the logarithms were invented by Napier and Kepler. Unfortunately, Kepler had no logarithm tables when he was faced with the laborious task of deriving his planet laws from Tycho Brahes measurements. Or the current…
reviewed by: Thomas Ernst
Mathematics Subject Classification ID: 33D15
zbMATH DE Number: 2181961
zbMATH Keywords: Jackson
zbMATH Keywords: basic hypergeometric series
zbMATH Keywords: \(q\)-analogue of Clausen's formula
MaRDI profile type: Publication
full work available at URL: j.disc.2005.03.001
OpenAlex ID: W2109996873
cites work: On the q-analog of Kummer's theorem and applications
cites work: On q-Analogues of the Watson and Whipple Summations
cites work: A NOTE ON CERTAIN q-IDENTITIES
cites work: ON THE ANALOGUE OF DIXON'S THEOREM FOR BILATERAL BASIC HYPERGEOMETRIC SERIES
cites work: ON THE SUM OF A TERMINATING 2F2(1)
cites work: Some formulas of F. H. Jackson
cites work: Parameter augmentation for basic hypergeometric series. II
cites work: Q4400925
cites work: \(q\)-identities from Lagrange and Newton interpolation
cites work: q-Extensions of Clausen’s Formula and of the Inequalities Used by de Branges in His Proof of the Bieberbach, Robertson, and Milin Conjectures
cites work: A Nonterminating q-Clausen Formula and Some Related Product Formulas
cites work: Q4040884
cites work: A combinatorial proof of a symmetric \(q\)-Pfaff-Saalschütz identity
cites work: THE qθ EQUATIONS WHOSE SOLUTIONS ARE PRODUCTS OF SOLUTIONS OF qθ EQUATIONS OF LOWER ORDER
cites work: CERTAIN q-IDENTITIES
cites work: Some Transformations of Basic Hypergeometric Functions. Part II
cites work: q-Series Identities and Reducibility of Basic Double Hypergeometric Functions
cites work: Some operator identities and \(q\)-series transformation formulas
cites work: The Basic Analogues of Identities of the Cayley-Orr Type
cites work: On some \(q\)-identities related to divisor functions
DOI: 10.1016/J.DISC.2005.03.001
description: scientific article; zbMATH DE number 2181961