The quantum group of plane motions and basic Bessel functions
The author considers matrix elements of irreducible representations of the quantum group of plane motions determined in some weak sense (generalized matrix elements), that is, as functionals on the quantized universal enveloping algebra. They have relative bi-invariance properties with respect to infinitesimally characterized quantum subgroups. This relative bi-invariance is the consequence of the fact that we can determine a basis of eigenvectors for different eigenvalues of a certain selfadjoint operator in \(l^2 (\mathbb{Z})\). The eigenvectors are described in terms of Jackson's \(q\)-Bessel functions. The author finds Hansen- Lommel orthogonality relations for these \(q\)-Bessel functions up to a scalar as a consequence of relative bi-invariance. The action of the comultiplication on generalized matrix elements is investigated. From this action the author derives a \(q\)-analogue of Graf's addition theorem for \(q\)-Bessel functions which are expressible as a \(q\)-hypergeometric \({}_2 \varphi_1\)-series. A sketch is given showing how one can formally interpret the orthogonality relations for the \(q\)-Jacobi functions as Hankel type orthogonality relations.
\(q\)-analogue of Graf's addition theorem
scientific article; zbMATH DE number 792825
scientific article