Euler transformation formula for multiple basic hypergeometric series of type \(A\) and some applications
In this remarkable paper, the author establishes a rather general transformation formula between two basic hypergeometric series associated to root systems of type \(A\) which are \textit{of different dimensions}. This is remarkable because, up to this date, there do not appear many identities of this type in the literature. The only other examples that the reviewer is aware of are [\textit{I.~M.~Gessel} and the reviewer, Trans. Am. Math. Soc. 349, 429--479 (1997; Zbl 0865.05003); Sec.~8], the reviewer, in: [\(q\)-Series with Applications to Combinatorics, Number Theory, and Physics, Urbana-Champaign, Oct.~26--28, 2000, \textit{B.~C.~Berndt, K.~Ono} (eds.), Contemporary Mathematics. 291. Providence, RI: American Mathematical Society (AMS), 153--161 (2001; Zbl 0980.00024); Conjecture], [\textit{H.~Rosengren}, J. Math. Anal. Appl. 281, 332--345 (2003; Zbl 1032.33013), and Constructive Approximation 20, 525--548 (2004; Zbl 1077.33031)]. The author's formula reduces to one of Heine's transformation formulae between two \(_2\phi_1\)-series in its simplest case. The proof of the formula is based on results from the theory of Macdonald polynomials. The author provides numerous applications of his formula, such as proofs of Pfaff--Saalschütz and Gauß summation formulae for multiple basic hypergeometric series of type \(A_n\) (some of them having been found earlier by Milne), multiple series generalizations of Watson's transformation formula between series of different dimensions, a transformation formula between series of different dimensions generalizing at the same time Bailey's transformation formula between two very-well-poised \(_{10}\phi_9\)-series and Jackson's summation formula for a very-well-poised \(_8\phi_7\)-series, and a transformation formula between series of different dimensions which reduces to Sears' transformation formula between two balanced \(_4\phi_3\)-series in its simplest case. Reviewer's remark: H.~Rosengren [``New transformations for elliptic hypergeometric series on the root system \(A_n\),'' preprint \url{arXiv:math.CA/0305379}] has generalized the main formula of the paper under review to elliptic hypergeometric series. In a special case, this has also been done by the author and \textit{M.~Noumi} [Indag. Math., New Ser. 14, 395--421 (2003; Zbl 1051.33009)].
basic hypergeometric series associated to root systems
scientific article; zbMATH DE number 2093004
scientific article