Central factorial numbers associated with sequences of polynomials

TL;DR

This paper generalizes central factorial numbers to any polynomial sequence using umbral calculus, revealing their orthogonality and inverse relations, and illustrating these properties with various examples.

Contribution

It introduces a unified framework for central factorial numbers associated with polynomial sequences, extending their properties and applications.

Findings

Central factorial numbers associated with P have orthogonality properties.

Inverse relations are established for these numbers.

Applications to various polynomial sequences demonstrate the theory's versatility.

Abstract

Many important special numbers appear in the expansions of some polynomials in terms of central factorials and vice versa, for example central factorial numbers, degenerate central factorial numbers, and central Lah numbers which are recently introduced. Here we generalize this to any sequence of polynomials. Let P be the space of a sequence of polynomials such that the degree of the polynomials less than n . The aim of this paper is to study the central factorial numbers of the second associated with P and of the first kind associated with P, in a unified and systematic way with the help of umbral calculus technique. The central factorial numbers associated with P enjoy orthogonality and inverse relations. We illustrate our results with many examples and obtain interesting orthogonality and inverse relations by applying such relations for the central factorial numbers associated with P to each of our examples.