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

Author

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  • Feng Qi

    (School of Mathematics and Informatics, Henan Polytechnic University, Jiaozuo 454010, China
    17709 Sabal Court, Dallas, TX 75252-8024, USA
    School of Mathematics and Physics, Hulunbuir University, Hailar 021008, China)

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.

Suggested Citation

  • Feng Qi, 2025. "Series and Connections Among Central Factorial Numbers, Stirling Numbers, Inverse of Vandermonde Matrix, and Normalized Remainders of Maclaurin Series Expansions †," Mathematics, MDPI, vol. 13(2), pages 1-52, January.
    • Handle: RePEc:gam:jmathe:v:13:y:2025:i:2:p:223-:d:1564532
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