A q-analogue of Appell's \(F_ 1\) function and some quadratic transformation formulas for non-terminating basic hypergeometric series
A q-extension of Appell's integral representation of the double series \(F_ 1\) is given using the four parameter q-beta kernel times two \({}_ 1\phi_ 0's\). This integral is evaluated as the sum of three very well poised \(_{10}\phi_ 9's\). One limit as \(q\to 1\) gives the sum of three very well poised \({}_ 9F_ 8's\) as an integral. I do not understand why these results exist, but they are very interesting. Some interesting corollaries are quadratic transformations between very well poised \(_{10}\phi_ 9's\) and balanced \({}_ 5\phi_ 4's\).
scientific article; zbMATH DE number 3955415