An example of algebraization of analysis and Fibonacci cobweb poset characterization

TL;DR

This paper explores the algebraization of analysis through $ ext{Fibonomial}$ operator calculus and characterizes the Fibonacci Cobweb poset as a DAG and oDAG, linking combinatorics and algebraic calculus.

Contribution

It introduces the Finite Fibonomial Operator Calculus as a special case of $ ext{psi}$-extended Rota's calculus and characterizes the Fibonacci Cobweb poset as a DAG and oDAG.

Findings

Finite Fibonomial Operator Calculus detailed as a $ ext{psi}$-extended Rota calculus

Fibonacci Cobweb poset characterized as DAG and oDAG

Construction of a dim 2 poset with Hasse diagram matching the digraph of P

Abstract

In recent Kwasniewski's papers inspired by O. V. Viskov it was shown that the $\psi$-calculus in parts appears to be almost automatic, natural extension of classical operator calculus of Rota - Mullin or equivalently - of umbral calculus of Roman and Rota. At the same time this calculus is an example of the algebraization of the analysis - here restricted to the algebra of polynomials. The first part of the article is the review of the recent author's contribution. The main definitions and theorems of Finite Fibonomial Operator Calculus which is a special case of $\psi$-extented Rota's finite operator calculus are presented there. In the second part the characterization of Fibonacci Cobweb poset P as DAG and oDAG is given. The dim 2 poset such that its Hasse diagram coincide with digraf of P is constructed.