\(q\)-analogue of the Watanabe unitary transform associated to the \(q\)-continuous Gegenbauer polynomials (Q5928847)

For \(\lambda >0\), let \(G_\lambda\) denote the \(L^2\)-space on \(]-1,1[\) for which the Gegenbauer polynomials \(\{C_n^\lambda: n\geq 0\}\) form a complete orthogonal system. \textit{S. Watanabe} exhibited in [Tokyo J. Math. 13, No. 2, 421-427 (1990; Zbl 0735.46012)] an explicit unitary integral transform from \(G_\lambda\) onto a certain Fock-type Hilbert space \(H^{2,\lambda}(D)\) of analytic functions on the unit disk \(D\). The paper under review constructs for \(0<q<1\) a \(q\)-analogue of the above transform for the \(L^2\)-space \(G_{q,\lambda}\) on \(]-1,1[\) for which a complete orthogonal system consists of the \(q\)-continuous Gegenbauer polynomials \(\{C^\lambda_n(\cdot;q): n\geq 0\}\). A \(q\)-Fock space of analytic functions \(H^{2,\lambda}_q\) is defined so that \(H^{2,\lambda}\) corresponds to the limit \(q \rightarrow 1^-\). A formula for the reproducing kernel of \(H^{2,\lambda}_q\) is also proven. Then an integral operator with explicitly written kernel is shown to give the required isometry from \(G_{q,\lambda}\) onto \(H^{2,\lambda}_q\).