Duality, polarity and convolution in umbral calculus

2025/05/02 by Grivaux, Julien
#05A40 #12D10 #39A70 #Complex Variables (math.CV) #FOS: Mathematics

paper · doi:10.48550/arxiv.2505.01274

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.

Citations

• Connecting the q-Multiplicative Convolution and the Finite Difference Convolution
• Finite free convolutions of polynomials
• Elements of Polya-Schur theory in finite difference setting
• Homogeneous multivariate polynomials with the half-plane property

Related

• Dual Numbers and Operational Umbral Methods
• Recent contributions to the calculus of finite differences: a survey
• The classical umbral calculus: Sheffer sequences
• Umbral Calculus, a Different Mathematical Language
• Operational Umbral Calculus