Polynomials in algebraic analysis

TL;DR

This paper generalizes the concept of polynomials within algebraic analysis for right invertible operators, extending classical and quantum calculus Taylor formulas to multiple variables.

Contribution

It introduces a unified framework for defining and analyzing polynomials associated with families of right invertible operators, broadening the scope of algebraic and quantum calculus.

Findings

Generalized polynomials for multiple variables

Extended algebraic Taylor formulas

Unified approach for algebraic and quantum calculus

Abstract

The concept of polynomials in the sense of algebraic analysis, for a single right invertible linear operator, was introduced and studied originally by D. Przeworska-Rolewicz \cite{DPR}. One of the elegant results corresponding with that notion is a purely algebraic version of the Taylor formula, being a generalization of its usual counterpart, well known for functions of one variable. In quantum calculus there are some specific discrete derivations analyzed, which are right invertible linear operators \cite{kac}. Hence, with such quantum derivations one can associate the corresponding concept of algebraic polynomials and consequently the quantum calculus version of Taylor formula \cite{MULT2}. In the present paper we define and analyze, in the sense of algebraic analysis, polynomials corresponding with a given family of right invertible operators. Within this approach we generalize the usual polynomials of several variables.