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Generalizations of Watson’s theorem on the sum of a \(_ 3F_ 2\). (English) Zbl 0793.33005

The authors obtain the sums of the 25 series \[ _ 3F_ 2 \left[ \left. \begin{matrix} a,\;b,\;c \\ {1 \over 2} (a+b+1+i),\;2c+j \end{matrix} \right | 1 \right],\quad i,j \in \{-2,-1,0,1,2\}, \] using Rainville’s relations between contiguous functions and the instances already known, in particular, Watson’s theorem \((i=0,\;j=0)\). Furthermore, 50 sums of terminating series are given in which \((a,b)\) are replaced by \((a+2n,\;- 2n)\) and \((a+2n+1,\;-2n-1)\), respectively. Finally, they give 10 sums in the limiting cases \(i=2\), \(b \to a-1\), and \(i=1\), \(b \to a\). Each group of results is arranged as a master formula involving certain coefficients that are given in adjacent tables.

MSC:

33C20 Generalized hypergeometric series, \({}_pF_q\)
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