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

TL;DR

This paper corrects previous formulas for Wick products of $q$-Charlier polynomials, explores poset structures governing monomials and Wick polynomials, and derives new inversion and product formulas for various polynomial families.

Contribution

It identifies errors in prior product formulas, introduces poset-based methods for Wick polynomial relations, and provides new inversion and product formulas for multiple polynomial families.

Findings

Corrected product and linearization formulas for $q$-Charlier polynomials.

Computed M"obius functions for related posets.

Derived new inversion and product formulas for Hermite, Chebyshev, Charlier, and Laguerre polynomials.

Abstract

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