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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.

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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