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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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    References listed on IDEAS

    as
    1. Feng Qi, 2024. "Absolute Monotonicity of Normalized Tail of Power Series Expansion of Exponential Function," Mathematics, MDPI, vol. 12(18), pages 1-11, September.
    2. Da-Wei Niu & Feng Qi, 2024. "Monotonicity Results of Ratios between Normalized Tails of Maclaurin Power Series Expansions of Sine and Cosine," Mathematics, MDPI, vol. 12(12), pages 1-20, June.
    3. Xin-Le Liu & Hai-Xia Long & Feng Qi, 2023. "A Series Expansion of a Logarithmic Expression and a Decreasing Property of the Ratio of Two Logarithmic Expressions Containing Sine," Mathematics, MDPI, vol. 11(14), pages 1-12, July.
    4. Jonathan M. Borwein & Marc Chamberland, 2007. "Integer Powers of Arcsin," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 2007, pages 1-10, May.
    Full references (including those not matched with items on IDEAS)

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