Operator identities and several \(U(n+1)\) generalizations of the Kalnins--Miller transformations
Multiple basic hypergeometric series associated to the unitary group \(U(n+1)\) have been studied by many authors. In particular, in \textit{S. C. Milne}, [``Balanced \(_ 3\phi_ 2\) summation theorems for \(U(n)\) basic hypergeometric series.'' Adv. Math. 131, No.1, 93-187 (1997; Zbl 0886.33014)], among many other things, several \(U(n+1)\) generalizations of the \(q\)-Chu-Vandermonde summation formula were obtained. In the paper under review, starting fom those summations formulas, the author proves several \(U(n+1)\) generalizations of the Kalnins-Miller transformation (this was introduced, in the one dimensional case, in \textit{E. G. Kalnins, W. Miller, jun.} [``\(q\)-series and orthogonal polynomials associated with Barnes' first lemma.'' SIAM J. Math. Anal. 19, No.5, 1216-1231 (1988; Zbl 0652.33007)] and studied in \textit{Z.-G. Liu}, [``Some operator identities and \(q\)-series transformation formulas.'' Discrete Math. 265, No.1-3, 119-139 (2003; Zbl 1021.05010)]. The author makes also use of the \(q\)-exponential operator technique, introduced in \textit{W. Y. C. Chen, Z.-G. Liu}, [``Parameter augmentation for basic hypergeometric series. I.'' Mathematical essays in honor of Gian-Carlo Rota's 65th birthday. Boston, MA: Birkhäuser. Prog. Math. 161, 111-129 (1998; Zbl 0901.33008)] and \textit{W. Y. C. Chen, Z.-G. Liu}, [``Parameter augmentation for basic hypergeometric series. II.'' J. Comb. Theory, Ser. A 80, No.2, 175-195 (1997; Zbl 0901.33009)].
scientific article; zbMATH DE number 5073013