  The subtraction or addition of a prescribed number of photons to a field mode
does not, in general, simply shift the probability distribution by the number
of subtracted or added photons. Subtraction of a photon from an initial
coherent state, for example, leaves the photon statistics unchanged and the
same process applied to an initial thermal state increases the mean photon
number. We present a detailed analysis of the effects of the initial photon
statistics on those of the state from which the photons have been subtracted or
to which they have been added. Our approach is based on two closely-related
moment generating functions, one well-established and one that we introduce.
