On the quantum group and quantum algebra approach to \(q\)-special functions (Q2366253)

It has been shown recently that quantum groups and algebras provide an algebraic setting for \(q\)-special functions. There are two different interpretations, in one case, one proceeds in close analogy with Lie theory and considers the elements of the quantum algebra \(U_ q({\mathfrak G})\) that are obtained upon replacing the exponential map from the algebra \(\mathfrak G\) into the group \(G\) by \(q\)-exponential mapping. In the other case, one considers rather the Hopf algebra \(A(G_ q)\) associated to the quantum group \(G_ q\) which is a subalgebra of the dual of \(U_ q({\mathfrak G})\), generated by the coordinates of \(G_ q\). Here in this paper the authors explain the connection between these two approaches by using as an example the case where \(U_ q((\text{sl}(2))\) is the basic structure.