An example of algebraization of analysis and Fibonacci cobweb poset characterization

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.

💡 Research Summary

The paper is divided into two complementary parts, both illustrating the theme of “algebraization of analysis” within the realm of polynomial algebra.

In the first part the author revisits the ψ‑calculus, a framework originally introduced by Viskov and later developed by Kwasniewski, which generalizes Rota‑Mullin’s finite operator calculus and the umbral calculus of Roman and Rota. The ψ‑function is a sequence‑valued parameter that allows one to define a ψ‑difference operator Δ_ψ and a ψ‑shift (or raising) operator E_ψ. When the underlying sequence is chosen to be the Fibonacci numbers {F_n}, the operators become Δ_F and E_F, and the associated binomial coefficients turn into the so‑called Fibonomial coefficients
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