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arxiv: 2604.02384 · v1 · submitted 2026-04-01 · 🧮 math.NT

General formulas for a class of Euler sums

David H Bailey , Ross McPhedran , Bruno Salvy This is my paper

Pith reviewed 2026-05-13 21:40 UTC · model grok-4.3

classification 🧮 math.NT
keywords Euler sumsharmonic numbersdigamma functionpolygamma functionsclosed-form expressionsrational functionspartial fraction decomposition
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The pith

An algorithm produces closed-form expressions in digamma and polygamma functions for infinite sums of the form R(k) times the kth harmonic number, where R is any rational function whose denominator degree exceeds its numerator degree by at

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops and describes an easy-to-implement algorithm that converts sums of the form sum R(k) H_k, with R a rational function satisfying the degree condition, into explicit expressions built from digamma and polygamma values. The same technique reduces more general Euler sums with products of linear factors raised to powers in the denominator to a partial-fraction decomposition followed by known evaluations. Explicit formulas are supplied for the one- and two-factor cases with small exponents and multipliers, together with related sums containing an extra polynomial factor in the numerator. These closed forms replace numerical summation by direct evaluation at finitely many special-function arguments.

Core claim

There exists an algorithm that, for any qualifying rational R, returns an explicit closed form for sum R(k) H_k in terms of digamma and polygamma functions; the general multi-linear-denominator case reduces directly to partial-fraction decomposition, and concrete formulas are obtained for denominators consisting of one or two linear terms raised to powers up to three with coefficients up to four.

What carries the argument

The reduction of the rational-harmonic sum to a linear combination of polygamma functions of arguments linear in the summation index, obtained by applying known generating-function identities after partial-fraction decomposition of R.

If this is right

  • Explicit closed forms become available for all one- and two-term linear denominators with exponents up to three and small multipliers.
  • Sums containing an extra polynomial factor k^q in the numerator are likewise reduced to polygamma expressions.
  • High-precision numerical checks become a practical method for verifying the derived formulas in specific instances.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same reduction strategy may extend mechanically to denominators with three or more distinct linear factors once partial fractions are computed.
  • Because the output is expressed solely in polygamma functions, the formulas are immediately usable inside computer-algebra systems that already implement those functions.
  • The method supplies a systematic way to generate identities that can later be used to evaluate finite truncations or to relate different families of Euler sums.

Load-bearing premise

That every sum of this class admits a closed expression built only from digamma and polygamma functions and that the algorithm always recovers that expression.

What would settle it

A concrete rational function R satisfying the degree condition for which the numerical value of the sum at high precision differs from the value predicted by the algorithm's output formula.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper presents an easy-to-implement algorithm for obtaining explicit closed-form evaluations, expressed via digamma and polygamma functions, of convergent Euler sums ∑ R(k) H_k where R(k) is a rational function with denominator degree at least two greater than the numerator degree. It further reduces the general multi-linear-denominator case ∑ H_k / ∏ (m_i k + n_i)^{p_i} to partial-fraction decomposition and supplies explicit formulas for one- and two-term denominators with p_i ≤ 3 and m_i ≤ 4, together with related sums such as ∑ k^q H_k / (m k + n)^p; all formulas are checked by high-precision numerical evaluation on concrete instances.

Significance. If the algorithm and reductions are correct, the work supplies a systematic, implementable route to closed forms for a broad, convergent class of Euler sums that arise repeatedly in analytic number theory. The explicit formulas for small parameters and the reduction to known polygamma identities constitute concrete, usable results; the high-precision numerical checks provide direct, falsifiable support for the claimed identities.

minor comments (3)
  1. [§3] §3 (algorithm description): the step that converts the rational function R(k) into a linear combination of polygamma sums is only sketched; an explicit statement of the residue or coefficient extraction rule used would make the procedure fully reproducible from the text alone.
  2. [Table 2] Table 2 (two-term explicit formulas): the entry for p1=2, p2=2 lists a combination of polygamma(1,·) and polygamma(2,·) terms whose coefficients depend on m1,m2,n1,n2; the paper does not state whether these coefficients remain valid when m1=m2 (repeated linear factor), which would require a separate limiting case.
  3. [§5] §5 (numerical verification): the manuscript states that formulas were checked to “very high precision” but does not record the working precision (e.g., 1000 decimal digits) or the software library employed; adding this datum would strengthen the reproducibility claim.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our work and the recommendation of minor revision. The referee's summary correctly captures the scope of the algorithm for closed forms of Euler sums via polygamma functions, the reduction to partial fractions, and the explicit formulas provided for small parameters, along with the high-precision numerical verifications.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper's derivation relies on partial-fraction decomposition of the rational function R(k) followed by direct application of standard polygamma identities for the resulting sums; these steps are independent of the target closed forms and do not reduce to fitted parameters, self-citations, or ansatzes introduced by the authors themselves. High-precision numerical checks serve only as post-derivation verification on concrete instances and do not enter the general algorithm. No load-bearing self-citation chain or self-definitional reduction is present in the stated procedure.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work builds on standard mathematical axioms and assumptions about the form of the closed expressions for the sums.

axioms (2)
  • standard math The digamma and polygamma functions satisfy certain recurrence and reflection relations used in the derivations.
    These are well-established properties in the theory of special functions.
  • domain assumption Closed-form expressions exist for the specified class of sums in terms of digamma and polygamma functions.
    The algorithm is presented to compute them, assuming they can be expressed this way.

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Reference graph

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