$$q$$-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:

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