Two results of the summations for hypergeometric series (Q2779031)
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scientific article; zbMATH DE number 1723847
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | Two results of the summations for hypergeometric series |
scientific article; zbMATH DE number 1723847 |
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2 March 2003
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hypergeometric series
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summation formulae
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algorithm
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0.7550122
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0.7505567
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0.7443984
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0.7401961
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Two results of the summations for hypergeometric series (English)
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By choosing special parameters, the author obtains two summation formulae of hypergeometric series \({_5 F_4}\) and \({_7 F_6}\) in terms of the combination of digamma functions. Here we use the standard notation that NEWLINE\[NEWLINE{_r F_s}= \left[{a_1,\cdots,a_r\atop b_1,\cdots, b_s}; z\right]=\sum_{n=0}^\infty{(a_1)_n\cdots(a_r)_n\over n! (b_1)_n\cdots (b_s)_n} z^n. NEWLINE\]NEWLINE We only quote the first result here: NEWLINE\[NEWLINE\begin{multlined} {_5 F_4}\left[{2+{2\over c}, 1, 1, 1+c, 1\atop 1+{c\over 2}, 2+c, 2+c, 2}; -1\right]=\\ {(c+1)^2\over 2(c+2)}\left({1\over 4}\left(\psi \left({1-c\over 2}\right)-\psi\left(1-{c\over 2}\right)+ \psi\left({c\over 2}\right)-\psi\left({1+c\over 2}\right)\right)^2+\psi^{(1)}(1+c)-\psi^{(1)}(-c)\right)^2.\end{multlined} NEWLINE\]NEWLINE The proof is based on results of digamma functions taken from two previous papers of the author quoted in the reference section. The reviewer was only able to local one of the papers of the author [J. Jiangxi Norm. Univ., Nat. Sci. Ed. 23, No. 1, 75-77 (1999; Zbl 0978.33011)]. There are typing mistakes in the acticle's title, page numbers of the references quoted.
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