Central factorial numbers associated with sequences of polynomials

• Published in: Mathematical methods in the applied sciences Vol. 46; no. 9; pp. 10348 - 10383
• Main Authors: San Kim, Dae, Kim, Taekyun
• Format: Journal Article
• Language: English
• Published: Freiburg Wiley Subscription Services, Inc 01.06.2023
• Subjects: associated with sequence of polynomials; central factorial numbers of the first kind; central factorial numbers of the second kind; Factorials; Orthogonality; Polynomials; umbral calculus
• ISSN: 0170-4214; 1099-1476
• DOI: 10.1002/mma.9127

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 P=pn(x)n=0∞$$ \boldsymbol{P}={\left\{{p}_n(x)\right\}}_{n=0}^{\infty } $$ such that deg  pn(x)=n,p0(x)=1$$ {p}_n(x)=n,{p}_0(x)=1 $$. The aim of this paper is to study the central factorial numbers of the second kind associated with any sequence of polynomials and of the first kind associated with any sequence of polynomials, in a unified and systematic way with the help of umbral calculus technique. The central factorial numbers associated with any sequence of polynomials 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 any sequence of polynomials to each of our examples.