Generating kernel for the boson realisation of symplectic algebras

The discussion of boson realisations of symplectic algebras requires as an essential ingredient an operator K needed for the passage from the Dyson to the Holstein-Primakoff realisation. In previous papers the matrix form of K² was derived through appropriate recursion relations. In the present analysis the authors show that K² can be determined in explicit and closed form as the overlap of coherent states of the symplectic group. The matrix elements of K² can then be obtained by expanding this overlap in terms of appropriate eigenstates.