Multidimensional matrix inversions and \(A_r\) and \(D_r\) basic hypergeometric series
This is a significant contribution to the theory of multiple basic hypergeometric series associated to root systems. This theory has been developed over the past twenty years by mainly Milne and Gustafson and co-workers. A spectacular breakthrough was achieved recently by \textit{S. C. Milne} and \textit{G. M. Lilly} [Bull. Am. Math. Soc., New Ser. 26, No. 2, 258-263 (1992; Zbl 0757.33008)] by extending the classical Bailey transform to the multidimensional case. Thus, \textit{S. C. Milne, G. M. Lilly} [Discrete Math. 139, No. 1-3, 319-346 (1995; Zbl 0870.33012)]; \textit{S. C. Milne} and \textit{J. W. Newcomb} [J. Comput. Appl. Math. 68, No. 1-2, 239-285 (1996; Zbl 0870.33013)] were able to derive new multidimensional generalizations of classical identities such as Jackson's very-well-poised \(_8\varphi_7\)-summation, Watson's \(_8\varphi_7\) to \(_4\varphi_3\) transformation, etc. As \textit{G. E. Andrews} [in: Proc. Symp. Pure Math., Am. Math. Soc., Columbus, Ohio 1978, Proc. Symp. Pure Math. 34, 1-24 (1979; Zbl 0403.33002)] showed, the Bailey transformation is basically equivalent to a particular matrix inversion. On the other hand, this matrix inversion is just a special case of the reviewer's matrix inversion [Proc. Am. Math. Soc. 124, No. 1, 47-59 (1996; Zbl 0843.15005)]. The latter matrix inversion does have other specializations as well which are important in the theory of (one-dimensional) basic hypergeometric series, in particular the specialization that enabled \textit{G. Gasper} and \textit{M. Rahman} to derive many quadratic, cubic and quartic basic hypergeometric identities [see Chap. 3.6 in the book ``Basic hypergeometric series'' (1990; Zbl 0695.33001)]. The central result of the paper under review is an extension (actually two extensions) to the multivariate setting of the reviewer's matrix inversion. The author proves it by using the operator method of the reviewer from [Trans. Am. Math. Soc. 305, No. 2, 431-465 (1988; Zbl 0653.05007)]. With this versatile tool at his hand, the author is able to derive a new multidimensional extension (related to the root system \(D_r\)) of Jackson's \(_8\varphi_7\)-summation [discovered independently by \textit{G. Bhatnagar}, Ramanujan J. 3, No. 2, 175-203 (1999; Zbl 0930.33012)], extensions of quadratic basic hypergeometric summation formulas by Gessel and Stanton, an extension of a cubic basic hypergeometric summation formula by Gasper and Rahman, and, in joint work with \textit{G. Bhatnagar} [Constructive Approximation 14, 531-567 (1998)], new multidimensional extensions of Bailey's very-well-poised \(_{10}\varphi_9\)-transformation formula.
scientific article; zbMATH DE number 1185667