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$$q$$-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.

##### 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
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##### 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physics, and computer algebra, () · Zbl 0594.33001 [3] Askey, R., Beta integrals in Ramanujan’s papers, his unpublished work and further examples, () · Zbl 0648.33001 [4] Floreanini, R.; Lapointe, L.; Vinet, L., A quantum algebra approach to basic multivariate special functions,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, (), 85-100 · Zbl 0807.33013 [13] Gasper, G.; Rahman, M., Basic hypergeometric series, (1990), 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 New York · Zbl 0174.10502 [19] Prudnikov, A.P.; Brychkov, Yu.A.; Marichev, O.I., () [20] Vilenkin, N.Ya., Special functions and the theory of group representations, () · Zbl 0172.18404 [21] Vilenkin, N.Ya.; Klimyk, A.U., Representation of Lie groups and special functions, (1991), Kluwer Dordrecht · Zbl 0742.22001
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