Inversion techniques and combinatorial identities: Balanced hypergeometric series.
type: Item
instance of: scholarly article
title: Inversion techniques and combinatorial identities: Balanced hypergeometric series.
zbMATH Open document ID: 1038.33002
published in: Rocky Mountain Journal of Mathematics
publication date: 2003-12-03T00:00:00Z
full work available at URL: CONT32-2.html
review text: After constructing the duplicate form of the Gould-Hsu inversion theorem, the author applies it to the derivation of summation theorems for terminating hypergeometric series with variable unity. The first type to be considered is \[ {_{5}F_{4}}\left[ \begin{matrix} -\delta -2n,\,a+x+n,\,b+x,\,z,\,\frac{1}{2}+x-z; \\ d+2z,e+2x-2z,\frac{1}{2}\left( c+x-n\right) ,\frac{1}{2}\left( 1+c+x-n\right) -\delta ; \end{matrix} \;1\right] , \] where \(x,z\in \mathbb{C}\), and \(n\in \mathbb{N}_{0}\); the remaining symbols denote constants. For a number of sextuples \(( a,b,c,d,e,\delta),\) where \(a,b,c,d,e\) are small integers and \(\delta\) equals \(0\) or \(1\), the author obtains summation formulas (a total of 31), most of which involve Saalschützian \({_{5}F_{4}}[1] \) 's. Some summation theorems due to Andrews and to Stanton are recovered. One of Gessel \& Stanton's ``strange'' evaluations (a terminating \( {_{7}F_{6}}[1]\)) is helpful. Next, by introduction of additional parameters \(\binom {\alpha ,\beta ,\gamma} {1+\alpha ,1+\beta ,1+\gamma}\) the author obtains an \({_{8}F_{7}}[1],\) for which two summation theorems are then established. Finally, the author discusses \(q\)-extension and reversal.
Mathematics Subject Classification ID: 33C20
Mathematics Subject Classification ID: 05A19
zbMATH DE Number: 2012108
zbMATH Keywords: duplicate inverse series relations
author: Wenchang Chu
reviewed by: Per W. Karlsson
MaRDI profile type: Publication
OpenAlex ID: W2049637968
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DOI: 10.1216/RMJM/1030539687
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description: scientific article; zbMATH DE number 2012108