ARBITRARY-ORDER HERMITE GENERATING FUNCTIONS FOR COHERENT AND SQUEEZED STATES

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For use in calculating higher-order coherent- and squeezed- state quantities, we derive generalized generating functions for the Hermite polynomials. They are given by ∑ₙ₌₀∞zjn+kHjn+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 [aʲ(d/dx)ʲ] on well-behaved functions and apply them to obtain other sums.