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General formulas for a class of Euler sums

David H Bailey, Ross McPhedran, Bruno Salvy

Abstract

Let $H_k = 1 + 1/2 + 1/3 + \cdots + 1/k$ denote the $k$th harmonic number. We present an easy-to-implement algorithm for the computation of explicit closed-form evaluations, in terms of the digamma and polygamma functions, for Euler sums of the form \begin{align} \sum_{k=1}^\infty R(k) H_k, \end{align} where $R(k)$ is a rational function (quotient of two polynomials) whose denominator degree is at least two larger than the numerator degree. We apply the same method to show how the computation of a general formula for Euler sums of the form \begin{align*} \sum_{k=1}^\infty \frac{H_k}{(m_1 k + n_1)^{p_1} (m_2 k + n_2)^{p_2} \cdots (m_r k + n_r)^{p_r}} \end{align*} reduces to partial fraction decomposition. We present explicit formulae for sums with one or two terms in the denominator, with powers $p_i$ ranging up to 3, and with multipliers $m_i$ ranging up to 4. We also include results for related Euler sums such as \begin{align*} \sum_{k=1}^\infty \frac{k^q H_k}{(m k + n)^p}. \end{align*} Computation of Euler sums directly to very high precision enables us to rigorously check the above-mentioned formulas in many specific cases.

General formulas for a class of Euler sums

Abstract

Let denote the th harmonic number. We present an easy-to-implement algorithm for the computation of explicit closed-form evaluations, in terms of the digamma and polygamma functions, for Euler sums of the form \begin{align} \sum_{k=1}^\infty R(k) H_k, \end{align} where is a rational function (quotient of two polynomials) whose denominator degree is at least two larger than the numerator degree. We apply the same method to show how the computation of a general formula for Euler sums of the form \begin{align*} \sum_{k=1}^\infty \frac{H_k}{(m_1 k + n_1)^{p_1} (m_2 k + n_2)^{p_2} \cdots (m_r k + n_r)^{p_r}} \end{align*} reduces to partial fraction decomposition. We present explicit formulae for sums with one or two terms in the denominator, with powers ranging up to 3, and with multipliers ranging up to 4. We also include results for related Euler sums such as \begin{align*} \sum_{k=1}^\infty \frac{k^q H_k}{(m k + n)^p}. \end{align*} Computation of Euler sums directly to very high precision enables us to rigorously check the above-mentioned formulas in many specific cases.

Paper Structure

This paper contains 16 sections, 6 theorems, 51 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. Formulas by residue computation
  3. Background on the psi function
  4. Harmonic sums by residue computation
  5. Algorithm
  6. General formulas
  7. Simple poles
  8. A unique pole of higher order
  9. Mixed denominators
  10. Numerical computation of mixed-denominator Euler sums
  11. Computing digamma and polygamma
  12. Conclusion
  13. Maple code for Algorithm \ref{['algo:algo1']}
  14. Catalog of formulas
  15. Theorem \ref{['thm:thm2']} (Corollary \ref{['cor:cor1']}) formulas
  16. ...and 1 more sections

Key Result

Proposition 1

Let $R(z)$ be a rational function (quotient of two polynomials) that is $O(z^{-2})$ at infinity with no poles at integers except possibly at 0. Then where $\Res [Q(s)]_{s=\alpha}$ denotes the residue, in the complex analysis sense, of the function $Q(s)$ at $s = \alpha$. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (17)

  • Proposition 1
  • proof
  • Example 1
  • Example 2
  • Example 3
  • Example 4
  • Theorem 1
  • proof
  • Example 5
  • Theorem 2
  • ...and 7 more