A hypergeometric hierarchy for the Andrews evaluations
The first part of the paper is concerned with hypergeometric series \({_{r+1}F_r}[1]\) that are 1-balanced (Saalschützian) and terminating, with \(r+3\) free parameters, \(r\in \{4,5,6,7\}\). \textit{G. E. Andrews} [Discrete Math. 193, No. 1-3, 43-60 (1998)] considered the case \(r=4\); and he gave 30 evaluations of such \({_5F_4}[1]\)'s for special choices of the parameters. The author establishes further evaluations and lists 2- formulas in the case \(r=5\). Moreover, he succeeds in fitting all these evaluations into a hierarchy the top of which is a theorem giving one evaluation in the case \(r=7\). But suitable specialization of the parameters in this \({_8F_7}[1]\), the other evaluations emerge. In particular, 165 evaluations for the \(r=4\) case are obtainable. Of these, 145 are distinct, and all those obtained by Andrews are included. A list of these 145 items could not be given. The second part of the paper is concerned with hypergeometric transformations that are obtainable from the evaluations. Only a few examples of such transformations are given; each has the form \[ R(y){_4F_3}[4y(1- y)]+ S(y){_4F_3}\Biggl[{-y^2\over 4(1- y)}\Biggr]+ T(y){_4F_3} \Biggl[{-4y\over(1- y)^2}\Biggr], \] where \(R\), \(S\), \(T\) are elementary functions; parameters are not shown.
scientific article; zbMATH DE number 1251334