Identify the appropriate pairing for general Sheffer sequences (ϑ-duality)

Determine an explicit bilinear pairing on the vector space of polynomials of degree at most n that corresponds to the ϑ-based duality for general Sheffer sequences: given a translation-invariant delta operator ϑ associated with a Sheffer sequence (S_n) and the dual polynomials defined via exterior products of the curve θ_t = (S_0(t),…,S_n(t)) with ϑθ_t, …, ϑ^nθ_t, identify the pairing on polynomials that generalizes the classical polarity pairing (recovered in the Appell case ϑ = d/dt) and for which these dual polynomials form the dual basis, beyond the special finite-difference case ϑ = Δ.

Background

The paper constructs a duality for Vandermonde curves using Wronskians and shows it coincides (up to normalization/sign) with duality for the classical polarity pairing in the Appell case, yielding determinantal representations and connections to classical results.

The authors then attempt to extend this trinity—wronskian duality, polarity pairing, and Appell sequences—to the broader class of Sheffer sequences. For a Sheffer sequence, the associated translation-invariant delta operator ϑ (a formal series in derivatives) replaces the usual derivative, and a natural ϑ-based duality can be defined via exterior products of θ_t, ϑθ_t, …, ϑ^nθ_t.

However, while the duality construction extends, the corresponding bilinear pairing that should play the role of the polarity pairing is not identified in general. The paper provides such a pairing only in the special finite-difference case ϑ = Δ (using forward/backward differences), leaving the general ϑ case unresolved.