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

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.


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
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[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[3] Askey, R., Beta integrals in Ramanujan’s papers, his unpublished work and further examples, (Andrews, G. E.; etal., Ramanujan Revisited (1988), Academic Press: Academic Press New York) · Zbl 0648.33001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
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