Quasi-exactly solvable problems and the dual (q-)Hahn polynomials

A second-order differential (q-difference) eigenvalue equation is constructed whose solutions are generating functions of the dual (q-)Hahn polynomials. The fact is noticed that these generating functions are reduced to the (little q-)Jacobi polynomials, and implications of this for quasi-exactly solvable problems are studied. A connection with the Azbel-Hofstadter problem is indicated.Comment: 15 pages, LaTex; final version, presentation improved, title changed, to appear in J.Math.Phy

On the Clebsch-Gordan coefficients for the two-parameter quantum algebra SU(2)p,qSU(2)_{p,q}

We show that the Clebsch - Gordan coefficients for the $SU(2)_{p,q}$ - algebra depend on a single parameter Q = $\sqrt{pq}$ ,contrary to the explicit calculation of Smirnov and Wehrhahn [J.Phys.A 25 (1992),5563].Comment: 5 page

q-Analog of Gelfand-Graev Basis for the Noncompact Quantum Algebra U_q(u(n,1))

For the quantum algebra U_q(gl(n+1)) in its reduction on the subalgebra U_q(gl(n)) an explicit description of a Mickelsson-Zhelobenko reduction Z-algebra Z_q(gl(n+1),gl(n)) is given in terms of the generators and their defining relations. Using this Z-algebra we describe Hermitian irreducible representations of a discrete series for the noncompact quantum algebra U_q(u(n,1)) which is a real form of U_q(gl(n+1)), namely, an orthonormal Gelfand-Graev basis is constructed in an explicit form.Comment: Invited talk given by V.N.T. at XVIII International Colloquium "Integrable Systems and Quantum Symmetries", June 18--20, 2009, Prague, Czech Republi