Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2505.01274

Duality, polarity and convolution in umbral calculus

Julien Grivaux
View original source ↗Read source document ↗Export metadata ↓

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.

Explore related subjects

math.CV

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Julien Grivaux. 2025-11-07. Duality, polarity and convolution in umbral calculus. https://arxiv.org/abs/2505.01274

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Hilbert metric and quasiconformal mappings

We prove a functional identity between the Hilbert metric and the visual angle metric in the unit disk. The proof utilizes the Poincaré hyperbolic metric in terms of which both metrics can be expressed. This identity then yields sharp distortion results for quasiregular mappings and analytic functions, expressed in terms of the Hilbert metric. We also prove that Hilbert circles are, in fact, Euclidean ellipses. The proof makes use of computer algebra methods. In particular, Gröbner bases are used.

math.CV↗
arXiv2026

Cyclicity in Poletsky-Stessin Weighted Bergman Spaces

We study the cyclicity of polynomials in Poletsky-Stessin weighted Bergman spaces on various domains in $\mathbb{C}^2$, including the unit ball, the bidisk, and the complex ellipsoid. To this end, we introduce a natural extension of the parameter range for Poletsky-Stessin weighted Bergman spaces on complete Reinhardt domains, yielding a family of spaces that resemble Dirichlet-type spaces on the unit ball. We highlight the differences in the cyclicity behavior of polynomials in these spaces on the bidisk compared to those studied by Bénéteau et al. Finally, we propose several open problems concerning the structure of cyclic polynomials in these spaces.

math.CV↗
arXiv2026

Weak Solutions to the complex Monge-Ampère flows on compact Kähler manifolds : general measures on the right-hand side

We show the existence of a bounded solution to the Cauchy problem for the complex Monge-Ampère flow on a compact Kähler manifold, with the right-hand side of the form $dt \wedge dμ$ where $dμ$ is either a Monge-Ampère measure with a bounded potential or dominated by a Monge-Ampère measure with a Hölder continuous potential. For the second case, we also prove that for a given semi-positive big from $θ$, the $t$-slice of the solution is locally Hölder continuous on $\rm{Amp(θ)}$ for all $t \in (0, T)$. Next, we prove a comparison principle when $dμ$ is dominated by a Monge-Ampère measure of a bounded quasi-plurisubharmonic function, which implies the uniqueness of the solution.

math.CV↗