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Normalized Remainders: Definition, Origins of Ideas, Historic Backgrounds, Numerous Recent Results, Some Basic Properties, and Related Guesses and Problems

Feng Qi

Abstract

In this chapter, the author retrospects the ins and outs of the concept of the normalized remainders of the Maclaurin power series expansions of functions, reviews main results on normalized remainders of some elementary functions since 2023, explores the historic backgrounds of the normalized remainders in combinatorics and number theory on the Bernoulli numbers and polynomials, the Stirling numbers and polynomials, and their generalizations by Broder, Carlitz, and Howard, presents several basic properties of normalized remainders, and collects several guesses and problems posed while investigating normalized remainders of several elementary functions.

Normalized Remainders: Definition, Origins of Ideas, Historic Backgrounds, Numerous Recent Results, Some Basic Properties, and Related Guesses and Problems

Abstract

In this chapter, the author retrospects the ins and outs of the concept of the normalized remainders of the Maclaurin power series expansions of functions, reviews main results on normalized remainders of some elementary functions since 2023, explores the historic backgrounds of the normalized remainders in combinatorics and number theory on the Bernoulli numbers and polynomials, the Stirling numbers and polynomials, and their generalizations by Broder, Carlitz, and Howard, presents several basic properties of normalized remainders, and collects several guesses and problems posed while investigating normalized remainders of several elementary functions.

Paper Structure

This paper contains 44 sections, 4 theorems, 110 equations.

Table of Contents

  1. Definition of the normalized remainder
  2. Origins in mathematical analysis
  3. First origin related to tangent
  4. Second origin related to sine
  5. Third origin related to cosine
  6. Recent results of normalized remainders
  7. Normalized remainders of sine and cosine functions
  8. Normalized remainders of tangent function and its square
  9. Normalized remainders of exponential function
  10. Normalized remainders of inverse trigonometric functions
  11. Normalized remainders of a function involved in an integral representation of the reciprocal of the gamma function
  12. Normalized remainders of complete elliptic integrals
  13. Normalized remainder of power of inverse sine function
  14. Historic backgrounds in combinatorics and number theory
  15. Generating functions of Carlitz--Howard's generalizations of Bernoulli numbers and polynomials
  16. ...and 29 more sections

Key Result

Theorem 5.1

If $T_n[f(x),x_0]$ for some $n\in\mathbb{N}_0$ exists, then the normalized remainder $T_n[f(x),x_0]$ satisfies where $\alpha$ is a real number.

Theorems & Definitions (9)

  • Definition 1.1
  • Remark 4.1
  • Theorem 5.1: arcsine-liu-qi-mdpi.tex
  • Theorem 5.2: LiuXL-Arc-three.tex
  • Theorem 5.3: arcsine-liu-qi-mdpi.tex
  • Theorem 5.4
  • proof
  • Remark 6.1
  • Remark 7.1