Canonical Polynomial Sequences: Inverse Pairs

TL;DR

This paper explores the theory of canonical polynomial sequences generated by analytic functions, detailing their operator calculus, duality, and connections to classical polynomials like Gegenbauer, Bessel, and Touchard, using an operator calculus approach.

Contribution

It introduces a unified operator calculus framework for canonical polynomial sequences and their inverse pairs, connecting various classical polynomials through this formalism.

Findings

Established the duality between polynomial sequences and their inverse functions.

Connected Gegenbauer, Bessel, and Touchard polynomials within the operator calculus framework.

Provided examples illustrating the theory with explicit polynomial systems.

Abstract

For $V(z)$, analytic in a neighborhood of $0\in\mathbb{C}$, $V(0) = 0$, $V'(0)\ne0$, there is an associated sequence of polynomials, \textsl{canonical polynomials}, that is a generalized Appell sequence with lowering operator $V(d/dx)$. Correspondingly, the inverse function $Z(v)$ has an associated canonical polynomial sequence. The coefficients of the two sequences form two mutually inverse infinite matrices. We detail the operator calculus for the pair of systems, illustrating the duality of operators and variables between them. Examples are presented including a connection between Gegenbauer and Bessel polynomials. Touchard polynomials appear as well. Alternative settings would include an umbral calculus approach as well as the Riordan group. Here we use the approach through operator calculus.