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Quantum algebras and \(q\)-special functions. (English) Zbl 0773.33010
Summary: A quantum-algebraic framework for many \(q\)-special functions is provided. The two-dimensional Euclidean quantum algebra, \(\mathrm{sl}_q(2)\) and the \(q\)-oscillator algebra are considered. Realizations of these algebras in terms of operators acting on vector spaces of functions in one complex variable are given. Basic hypergeometric functions are shown to arise, in analogy with Lie theory, as matrix elements of certain operators. New generating functions for these \(q\)-special functions are obtained.

33D80 Connections of basic hypergeometric functions with quantum groups, Chevalley groups, \(p\)-adic groups, Hecke algebras, and related topics
17B37 Quantum groups (quantized enveloping algebras) and related deformations
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
81R12 Groups and algebras in quantum theory and relations with integrable systems
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