Spherical functions on the quantum group \(\text{SU}(1,1)\) and the \(q\)-analogue of the Mehler-Fock formula (Q803865)

The Plancherel decomposition formula for an arbitrary function on the quantum group \(\text{SU}(1,1)\) is obtained. This formula may be considered as the quantum analogue of the classical Mehler-Fock formula. It is pointed out that there exists a two-parameter family of quantum analogues of the group \(\text{SU}(1,1)\): to any pair of numbers \((h>0, \beta\in [0,1])\) there corresponds the Hopf algebra generated by matrix elements of the unitarizable representations which give contribution to the obtained Plancherel formula.