Vector coherent states with matrix moment problems

Canonical coherent states can be written as infinite series in powers of a single complex number $z$ and a positive integer $\rho(m)$. The requirement that these states realize a resolution of the identity typically results in a moment problem, where the moments form the positive sequence of real numbers $\{\rho(m)\}_{m=0}^\infty$. In this paper we obtain new classes of vector coherent states by simultaneously replacing the complex number $z$ and the moments $\rho(m)$ of the canonical coherent states by $n \times n$ matrices. Associated oscillator algebras are discussed with the aid of a generalized matrix factorial. Two physical examples are discussed. In the first example coherent states are obtained for the Jaynes-Cummings model in the weak coupling limit and some physical properties are discussed in terms of the constructed coherent states. In the second example coherent states are obtained for a conditionally exactly solvable supersymmetric radial harmonic oscillator.