Series and Connections Among Central Factorial Numbers, Stirling Numbers, Inverse of Vandermonde Matrix, and Normalized Remainders of Maclaurin Series Expansions †

Feng Qi ()
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Feng Qi: School of Mathematics and Informatics, Henan Polytechnic University, Jiaozuo 454010, China

Mathematics, 2025, vol. 13, issue 2, 1-52

Abstract: This paper presents an extensive investigation into several interrelated topics in mathematical analysis and number theory. The author revisits and builds upon known results regarding the Maclaurin power series expansions for a variety of functions and their normalized remainders, explores connections among central factorial numbers, the Stirling numbers, and specific matrix inverses, and derives several closed-form formulas and inequalities. Additionally, this paper reveals new insights into the properties of these mathematical objects, including logarithmic convexity, explicit expressions for certain quantities, and identities involving the Bell polynomials of the second kind.

Keywords: Vandermonde matrix; inverse matrix; Stirling number; Maclaurin power series expansion; normalized remainder; central factorial number; inverse sine function; logarithmic convexity; Vieta theorem; Bell polynomial of the second kind; complete elliptic integral of the second kind (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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