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It is shown that a SU(1,1) algebra may be used to provide a unifieddescription of the simple hamonic oscillator and the angular momentum algebrasand a class of other semi-infinite algebras. A normal ordered representation ofa Unitary operator $U$ constructed from the generators of a SU(1,1) algebra,which is a generalisation of the Baker- Campbell - Hausdorff relation for Liealgebras, is given. It is shown that the normal ordered representatiion of $U$may be used to calculate expectation values which are functions of theparameters used to construct the operator. The functions so constructed satisfycertain recurrence relations and the entire set of functions may be interpretedin terms of diagrams similar to the Pascal triangle for binomial coefficients.Coherent states, squeezed states and rotation matrices of the angular momentumalgebra emerge as special cases.

Recurrence relations and path representations of matrix elements of an SU(1,1) algebra