Elementary proofs of some \(q\)-identities of Jackson and Andrews--Jain
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 status of \(q\)-calculus is that of logarithms before the first big computer was built by Babbage.
scientific article; zbMATH DE number 2181961
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