Arbitrary-order Hermite generating functions for obtaining arbitrary-order coherent and squeezed states

Abstract For use in calculating higher-order coherent- and squeezed-state quantities, we derive generalized generating functions for the Hermite polynomials. They are given by Σ n=0 ∞ z jn+k H jn+k (x) (jn + k)! , for arbitrary integers j ≥ 1 and k ≥ 0. Along the way, the sums with the Hermite polynomials replaced by unity are also obtained. We also evaluate the action of the operators exp [c j ( d dx ) j ] on well-behaved functions and apply them to obtain other sums. Then the generating functions Σ n=0 ∞ z jn+k H jn+k (x)H jn+k (y) (jn + k)! are obtained. Finally, our techniques are used to define arbitrary-order coherent and squeezed states, i.e., those obtained from ladder operators of the form aj.