Duality, polarity and convolution in umbral calculus

TL;DR

This paper revisits the foundations of umbral calculus by introducing a matrix-based approach to binomial convolution, establishing duality and polarity concepts, and extending additive convolution with inversion formulas, leading to analogs of classical theorems.

Contribution

It introduces a new matrix realization of umbral calculus, constructs duality and polarity pairings, and extends convolution concepts with explicit inversion formulas.

Findings

Established an explicit matrix realization of binomial convolution.

Constructed an umbral duality of Wronskian type for rational curves.

Derived analogs of Grace and Walsh theorems for finite differences.

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.