Linear Transformations & the Multivariate Generating Function

This note examines linear combinations of multi-indexed sequences and derives the multivariate generating function of such a linear combination in terms of the original sequence's m.g.f. Applications include finding distributions and moments of non-negative discrete random variables conditioned on non-negative linear combinations of the original variables. Examples include independent Poisson r.v.'s and a $d$-variate multinomial distribution.