Stirling numbers associated with sequences of polynomials

TL;DR

This paper investigates Stirling numbers linked to polynomial sequences using umbral calculus, providing a unified framework and numerous examples that reveal interesting inverse relations.

Contribution

It introduces a systematic approach to study Stirling numbers associated with polynomial sequences via umbral calculus, unifying second and first kind analyses.

Findings

Derived new inverse relations involving Stirling numbers and polynomial sequences.

Provided numerous illustrative examples demonstrating the theoretical results.

Established a unified methodology for analyzing Stirling numbers in polynomial contexts.

Abstract

Let P be the set of the sequence of polynomials of degree n. The aim of this paper is to study the Stirling numbers of the second kind associated with P and of the first kind associated with P, in a unified and systematic way with the help of umbral calculus technique. Our results are illustrated with many examples which give rise to interesting inverse relations in each case.