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We give an example showing that the product and linearization formulas forthe Wick product versions of the $q$-Charlier polynomials in (Anshelevich 2004)are incorrect. Next, we observe that the relation between monomials and severalfamilies of Wick polynomials is governed by "incomplete" versions of familiarposets. We compute M\"obius functions for these posets, and prove a generalposet product formula. These provide new proofs and new inversion and productformulas for Wick product versions of Hermite, Chebyshev, Charlier, freeCharlier, and Laguerre polynomials. By different methods, we prove inversionformulas for the Wick product versions of the free Meixner polynomials.

Product formulas on posets, Wick products, and a correction for the $q$-Poisson process