Inversion techniques and combinatorial identities: Balanced hypergeometric series. (Q1415037)

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.