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A $q$-analog of a Whipple’s transformation for hypergeometric series in $U(n)$. (English) Zbl 0809.33010
Watson’s $q$-extension of Whipple’s transformation between a balanced series and a very well poised series is a central result in the study of basic hypergeometric functions in one variable. It is the minimal generalization of the Rogers-Ramanujan identities with no quadratic powers of $q$. For more than 15 years, one of the goals that Milne set for himself was to obtain a multivariate version of this identity which was not just an iterate of the one variable identity. He, along with a few others, has developed basic hpergeometric functions associated with $V(n)$. This paper builds on his earlier work and work of Robert Gustafson to find one such extension, and to point out some of the many interesting and beautiful special and limiting cases. Much more needs to be done to interpret limiting case in terms of partitions and other combinatorial objects, but it is clear there are gems here waiting to be discovered.

33D45Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.)
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