×

Operator identities and several \(U(n+1)\) generalizations of the Kalnins–Miller transformations. (English) Zbl 1113.33020

Multiple basic hypergeometric series associated to the unitary group \(U(n+1)\) have been studied by many authors. In particular, in 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 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 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 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 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)].

MSC:

33D67 Basic hypergeometric functions associated with root systems
05A30 \(q\)-calculus and related topics
33D15 Basic hypergeometric functions in one variable, \({}_r\phi_s\)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Andrews, G. E., \(q\)-Series: Their development and application in analysis, number theory, combinatorics, physics and computer algebra, CBMS Reg. Conf. Lectures Ser., vol. 66 (1986), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI · Zbl 0594.33001
[2] Chen, W. Y.C.; Liu, Z.-G., Parameter augmentation for basic hypergeometric series, II, J. Combin. Theory Ser. A, 80, 175-195 (1997) · Zbl 0901.33009
[3] Chen, W. Y.C.; Liu, Z-. G., Parameter augmentation for basic hypergeometric series, I, (Sagan, B. E.; Stanley, R. P., Mathematical Essays in Honor of Gian-Carlo Rota (1998), Birkhäuser: Birkhäuser Basel), 111-129 · Zbl 0901.33008
[4] Gasper, G. G.; Rahman, M., Basic Hypergeometric Series, Encyclopedia Math. Appl., vol. 96 (2004), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 1129.33005
[5] Kalnins, E. G.; Miller, W., \(q\)-Series and orthogonal polynomials associated with Barnes’ first lemma, SIAM J. Math. Anal., 19, 1216-1231 (1988) · Zbl 0652.33007
[6] Liu, Z.-G., Some operator identities and \(q\)-series transformation formulas, Discrete Math., 265, 119-139 (2003) · Zbl 1021.05010
[7] Milne, S. C., Balanced \({}_3\varphi_2\) summation theorems for \(U(n)\) basic hypergeometric series, Adv. Math., 131, 93-187 (1997) · Zbl 0886.33014
[8] Zhang, Zhizheng; Wang, Jun, Two operator identities and their applications to terminating basic hypergeometric series and \(q\)-integrals, J. Math. Anal. Appl., 312, 2, 653-665 (2005) · Zbl 1081.33032
[9] Zhizheng Zhang, Maixue Liu, Applications of operator identities to the multiple \(qq\); Zhizheng Zhang, Maixue Liu, Applications of operator identities to the multiple \(qq\) · Zbl 1095.05002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.