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The Trace Method for Cotangent Sums

Wiktor Ejsmont, Franz Lehner

Abstract

This paper presents a combinatorial study of sums of integer powers of the cotangent which is a popular theme in classical calculus. Our main tool the realization of cotangent values as eigenvalues of a simple self-adjoint matrix with integer matrix. We use the trace method to draw conclusions about integer values of the sums and expand generating functions to obtain explicit evaluations. It is remarkable that throughout the calculations the combinatorics are governed by the higher tangent and arctangent numbers exclusively. Finally we indicate a new approximation of the values of the Riemann zeta function at even integer arguments.

The Trace Method for Cotangent Sums

Abstract

This paper presents a combinatorial study of sums of integer powers of the cotangent which is a popular theme in classical calculus. Our main tool the realization of cotangent values as eigenvalues of a simple self-adjoint matrix with integer matrix. We use the trace method to draw conclusions about integer values of the sums and expand generating functions to obtain explicit evaluations. It is remarkable that throughout the calculations the combinatorics are governed by the higher tangent and arctangent numbers exclusively. Finally we indicate a new approximation of the values of the Riemann zeta function at even integer arguments.

Paper Structure

This paper contains 28 sections, 16 theorems, 134 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries on Linear Algebra and the Tangent function
  3. A matrix
  4. Formulas for $\tan(nx)$
  5. Formulas for $\tan(nx-\alpha)$
  6. Derivatives of $\tan$ and $\cot$
  7. Tangent and arctangent numbers
  8. Derivatives of $\arctan$
  9. Faà di Bruno's formula
  10. Trace formula
  11. Generating functions
  12. Generating function for cotangent sums
  13. A functional relation
  14. Combinatorial interpretation
  15. Dimension 2
  16. ...and 13 more sections

Key Result

Lemma 2.1

If $a=\cot \alpha$, then the characteristic polynomials $\chi_n(\alpha;\lambda)$ of the matrices satisfy the recurrence relation and have the following explicit expression (assuming $\lambda$ real). The eigenvalues are given by

Figures (3)

  • Figure 5.1: $T_{3,2}$ with corresponding weight of paths.
  • Figure 5.2: $T_{3,3}$ with corresponding weight of paths.
  • Figure 5.3: A circular forest; firstborns are marked with an extra circle

Theorems & Definitions (40)

  • Lemma 2.1
  • proof
  • Remark 2.2
  • Remark 2.3
  • Theorem 3.1
  • Example 3.2
  • proof
  • Corollary 3.3
  • proof
  • Proposition 4.1
  • ...and 30 more