Do any terms vanish in the recursively defined polynomials ψν?

Ascertain whether any monomial terms in the multivariate polynomials ψν(x1, …, xν), which are defined recursively via Bell polynomials and Newton’s identities in Theorem 4.1 and Theorem 4.3, cancel to zero in their expansions for some ν ≥ 1; specifically, determine whether any terms vanish in ψν for some ν.

Background

The paper introduces multivariate polynomials ψν(x1, …, xν) that arise in supercongruences for Wilson’s quotient and factorial modulo higher prime powers. These polynomials are computed recursively using Bell polynomials and Newton’s identities, and they have alternating signs in their coefficients.

Because of the recursive structure and sign alternations, the authors raise uncertainty about possible cancellation of monomials in ψν. This issue underlies the subsequent conjecture relating the number of monomials in ψν to a partition-based count.