Telescopic generalizations for two \(_3F_2\)-series identities
type: Item
instance of: scholarly article
zbMATH Open document ID: 1274.33007
author: Wenchang Chu
published in: Mathematica Slovaca
publication date: 2012-08-27T00:00:00Z
Mathematics Subject Classification ID: 33C20
zbMATH DE Number: 6072858
zbMATH Keywords: hypergeometric series
zbMATH Keywords: telescoping method
zbMATH Keywords: Thomae transformation
MaRDI profile type: Publication
full work available at URL: s12175-012-0038-5
OpenAlex ID: W2154962961
DOI: 10.2478/S12175-012-0038-5
Recommended article: Telescopic approach to a formula of \(_2F_1\)-series by Gosper and Ebisu
Recommended article: Q5120404
Recommended article: New series expansions of the \(_3F_2\) function
Recommended article: Multiple Generalizations of q-Series Identities and Related Formulas
Recommended article: Generalizations of classical summation theorems for the series2F1and3F2with applications
Recommended article: Q5442308
Recommended article: Whipple-type 3F2-series and summation formulae involving generalized harmonic numbers
Recommended article: A SUMMATION FORMULA FOR THE SERIES3F2DUE TO FOX AND ITS GENERALIZATIONS
Recommended article: ON SUMMATION THEOREMS FOR THE3F2(1) SERIES
Recommended article: Three Symmetric Double Series by Telescoping
title: Telescopic generalizations for two \(_3F_2\)-series identities
review text: Two nonterminating \(_3F_2(1)\)-series identities are derived: NEWLINE\[NEWLINE _3F_2\left [ \begin{alignedat}{2} 2,\;& 1+\alpha , & 1+\beta \gamma \\ & 2+\alpha \gamma , & 2+\beta \end{alignedat} \Big | 1\right ] = \frac {(1+\alpha \gamma)(1+\beta)}{(\alpha -\beta)(\gamma -1)} NEWLINE\]NEWLINE for \(\Re \{(\alpha -\beta)(\gamma -1)\}>0\) and NEWLINE\[NEWLINE _3F_2\left [\begin{alignedat}{2} 1+\alpha , \alpha \gamma , 1+\alpha \gamma -\beta \gamma \\ 2+\alpha \gamma , 1+\alpha \gamma -\beta \gamma +\beta \end{alignedat} \Big | 1 \right ]=(1+\alpha \gamma)\frac {\Gamma (1-\alpha +\beta)\Gamma (1+\alpha \gamma -\beta \gamma +\beta)}{\Gamma (1+\beta)\Gamma (1+(\alpha -\beta)(\gamma -1))} NEWLINE\]NEWLINE for \(\Re (1-\alpha +\beta)>0\), where \(\Gamma \) is Euler's gamma function.NEWLINENEWLINENEWLINENEWLINE The first of them confirms a conjecture of M. Milgram and the second one extends a couple of identities due to \textit{I. Gessel} and \textit{D. Stanton} [SIAM J. Math. Anal. 13, No. 2, 295--308 (1982; Zbl 0486.33003)].
description: scientific article; zbMATH DE number 6072858