Abstract
We survey summation theorems for generalized bibasic hypergeometric series found recently by Chu and perfected by Macdonald. These contain arbitrary sequences of parameters, and generalize bibasic summation theorems found by Gosper, Gasper, and Rahman. They also contain a host of matrix inversions, for infinite lower-triangular matrices, including those found by Gould, Hsu, Carlitz, and Krattenthaler. Further special cases of Krattenthaler's theorem are Andrews' matrix formulation of the Bailey Transform and Bressoud's and Gasper's inversions. We extend the telescoping arguments used for bibasic summation theorems to provideU(n+1) extensions of Gosper's and Gasper's bibasic summation theorems. Special cases includeU(n+1) extensions of Carlitz's and Gasper's matrix inversions. Further specializations correspond to theU(n+1) Bailey Transform andU(n+1) Bressoud matrix inversion found previously by Milne. Applications includeU(n+1) transformation and expansion formulas, and aU(n+1) extension of aq-analogue of Abel's Binomial Theorem. Thisq-analogue is different from those found by Jackson and Johnson.
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