×
Floreanini, Roberto; Vinet, Luc

\(q\)-gamma and \(q\)-beta functions in quantum algebra representation theory. (English) Zbl 0872.33010

J. Comput. Appl. Math. 68, No. 1-2, 57-68 (1996).
The authors consider a Hopf algebra \(\mathcal G_q\) which is generated by two generators. It is a \(q\)-deformation of the algebra of the infinitesimal affine transformations of the oriented line. By studying \(q\)-exponentials of the generators in some representations of \(\mathcal G_q\), the authors derive a number of identities featuring \(q\)-gamma and \(q\)-beta functions, and ordinary and bilateral basic hypergeometric series.
Reviewer: Ch.Krattenthaler (Wien)

Cited in 14 Documents

MSC:

33D05 \(q\)-gamma functions, \(q\)-beta functions and integrals
33D80 Connections of basic hypergeometric functions with quantum groups, Chevalley groups, \(p\)-adic groups, Hecke algebras, and related topics
16W30 Hopf algebras (associative rings and algebras) (MSC2000)
17B37 Quantum groups (quantized enveloping algebras) and related deformations

Keywords:

\(q\)-gamma function; \(q\)-beta function; quantum algebras; Hopf algebras; linear transformations of oriented line; addition formulas
PDF BibTeX XML Cite
\textit{R. Floreanini} and \textit{L. Vinet}, J. Comput. Appl. Math. 68, No. 1--2, 57--68 (1996; Zbl 0872.33010)
Full Text: DOI
  • logo of hbz
  • logo of WorldCat

References:

[1] Agarwal, A. K.; Kalnins, E. G.; Miller, W., Canonical equations and symmetry techniques for \(q\)-series, SIAM J. Math. Anal., 18, 1519-1538 (1987) · Zbl 0624.33005
[2] Andrews, G. E., \(q\)-Series: Their Development and Application in Analysis, Number Theory, Combinatorics, Physics, and Computer Algebra, (CBMS Regional Conference Lecture Series, 66 (1986), Amer. Math. Soc: Amer. Math. Soc Providence, RI) · Zbl 0594.33001
[3] Askey, R., Beta integrals in Ramanujan’s papers, his unpublished work and further examples, (Andrews, G. E.; etal., Ramanujan Revisited (1988), Academic Press: Academic Press New York) · Zbl 0648.33001
[4] Floreanini, R.; Lapointe, L.; Vinet, L., A quantum algebra approach to basic multivariate special functions, J. Phys. A, 27, 6781-6797 (1994) · Zbl 0864.33017
[5] Floreanini, R.; Vinet, L., \(q\)-Orthogonal polynomials and the oscillator quantum group, Lett. Math. Phys., 22, 45-54 (1991) · Zbl 0745.33008
[6] Floreanini, R.; Vinet, L., Addition formulas for \(q\)-Bessel functions, J. Math. Phys., 33, 2984-2988 (1992) · Zbl 0777.33011
[7] Floreanini, R.; Vinet, L., Using quantum algebras in \(q\)-speical function theory, Phys. Lett. A, 170, 21-28 (1992)
[8] Floreanini, R.; Vinet, L., Quantum algebras and \(q\)-special functions, Ann. Phys., 221, 53-70 (1993) · Zbl 0773.33010
[9] Floreanini, R.; Vinet, L., On the quantum group and quantum algebra approach to \(q\)-special functions, Lett. Math. Phys., 27, 179-190 (1993) · Zbl 0780.33012
[10] Floreanini, R.; Vinet, L., An algebraic interpretation of the \(q\)-hypergeometric functions, J. Group Theory Phys., 1, 1-10 (1993)
[11] Floreanini, R.; Vinet, L., Generalized \(q\)-Bessel functions, Can. J. Phys., 72, 345-354 (1994) · Zbl 1043.33500
[12] Floreanini, R.; Vinet, L., uq(sl(2)) and \(q- special\) functions, (Lie Algebras, Cohomology and New Applications to Quantum Mechanics. Lie Algebras, Cohomology and New Applications to Quantum Mechanics, Contemp. Math., 160 (1994), Amer. Math. Soc: Amer. Math. Soc Providence, RI), 85-100 · Zbl 0807.33013
[13] Gasper, G.; Rahman, M., Basic Hypergeometric Series (1990), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0695.33001
[14] Kalnins, E. G.; Manocha, H. L.; Miller, W., Models of \(q\)-algebra representations: I. Tensor products of special unitary and oscillator algebras, J. Math. Phys., 33, 2365-2383 (1992) · Zbl 0780.17014
[15] Kalnins, E. G.; Miller, W., Models of \(q\)-algebra representations: \(q\)-integral transform and “addition theorems”, J. Math. Phys., 35, 1951-1975 (1994) · Zbl 0817.17019
[16] Kalnins, E. G.; Miller, W.; Mukherjee, S., Models of \(q\)-algebra representations: Matrix elements of the \(q\)-oscillator algebra, J. Math. Phys., 34, 5333-5356 (1993) · Zbl 0795.17022
[17] Kalnins, E. G.; Miller, W.; Mukherjee, S., Models of \(q\)-algebra representations: The group of plane motions, SIAM J. Math. Anal., 25, 513-527 (1994) · Zbl 0805.33020
[18] Miller, W., Lie Theory and Special Functions (1968), Academic Press: Academic Press New York · Zbl 0174.10502
[19] Prudnikov, A. P.; Brychkov, Yu. A.; Marichev, O. I., (Integrals and Series, Vol. III (1990), Gordon and Breach: Gordon and Breach New York) · Zbl 0967.00503
[20] Vilenkin, N. Ya., Special Functions and the Theory of Group Representations, (Amer. Math. Soc. Transl. of Math. Monographs, 22 (1968), Amer. Math. Soc: Amer. Math. Soc Providence, RI) · Zbl 0172.18404
[21] Vilenkin, N. Ya.; Klimyk, A. U., Representation of Lie Groups and Special Functions (1991), Kluwer: Kluwer Dordrecht · Zbl 0742.22001
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.