A \(q\)-analogue of Capelli's identity for \(\text{GL}(2)\) (Q1908480)

In this paper, originally written in 1990, the authors derive a \(q\)-analogue of Capelli's identity for \(\text{GL}(2)\). In a note added in proof the authors refer to further results: \textit{M. Noumi}, \textit{T. Umeda}, and the second author worked out now to generalize a quantum analogue of Capelli's identity for \(\text{GL}(n)\) in the framework of quantum \(R\)-matrices [Duke Math. J. 76, 567--594 (1994; Zbl 0835.17013)]. This theory has been continuously well-developed from the representation theoretical point of view [Lett. Math. Phys. 34, 1--8 (1995; Zbl 0842.17020)].