Duality, polarity and convolution in umbral calculus

TL;DR

This work develops a finite-height, matrix-based framework for umbral calculus that centralizes binomial convolution and delta-operator theory via $\mathfrak{d}$ and its associated binomial sequence $\{S_k\}$. It introduces an umbral polarity pairing and an additive convolution $\boxplus_n^{\mathfrak{d}}$, derives explicit deviation polynomials $R_n$ as inverses of $S_n$, and establishes duality and Wronskian structures for Vandermonde and Sheffer curves, yielding explicit determinantal and convolution formulas. The authors extend Grace–Walsh type root-location results to the umbral finite-difference setting, showing that deviation roots align along a vertical line for $\Delta$ and providing umbral analogues of classical theorems. Collectively, the paper reveals rich algebro-geometric structures governing finite-difference polynomials, with concrete inversion, duality, and determinant identities that tie Appell, Sheffer, and binomial families together within a unified, finite-dimensional framework.

Abstract

In this paper, we revisit foundations of umbral calculus using a straightforward approach based on an explicit matrix realization of binomial convolution. We construct an umbral duality of Wronskian type for rational curves in echelon form, and connect it with an umbral version of the polarity pairing. Then we extend additive convolution to the umbral setting and provide an explicit inversion formula for the corresponding deviation terms. This enables us to derive analogs of Grace and Walsh representation theorems for finite differences.

Theorems & Definitions (96)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5: Grace theorem Grace1902, see Rahman
• Theorem 1.6: Walsh representation theorem Walsh, Rahman
• Theorem 1.7
• Theorem 1.8: Umbral Grace theorem for $\Delta$
• Theorem 1.9: Umbral Walsh theorem for $\Delta$
• Lemma 2.1