Arbitrary-Order Hermite Generating Functions for Coherent and Squeezed States
نویسندگان
چکیده
For use in calculating higher-order coherentand squeezedstate quantities, we derive generalized generating functions for the Hermite polynomials. They are given by ∑∞ n=0 z Hjn+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[a(d/dx)] on well-behaved functions and apply them to obtain other sums.
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nt - p h / 97 11 01 4 v 1 1 2 N ov 1 99 7 Reply to Comment on “ Generating Functions for Hermite Polynomials of Arbitrary Order ”
The results in the preceding comment are placed on a more general mathematical foundation. In the preceding comment [1], our previous results on arbitrary-order Hermite generating functions [2] were duplicated and extended. This was done by using a power-series expansion of the operator W = exp[−∂/4] to define the Hermite polynomials as Hn(x) = 2 Wx. We observe that this operator definition can...
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