Explicit Evaluations of Euler Sums Involving Harmonic Numbers with Rational Arguments

Technical Summary: Explicit Evaluations of Euler Sums Involving Harmonic Numbers with Rational Arguments

Problem Statement
The paper addresses the evaluation of a specific class of four-parameter Euler sums containing generalized harmonic numbers with rational arguments. The primary object of study is the infinite series:
$$\sum_{k=1}^{\infty} \frac{a^k H^{(p)}_{k/n}}{k^q}$$
where $p, n \in \mathbb{Z}_{\ge 1}$, $q \in \mathbb{Z}_{\ge 2}$, $a = \pm 1$, and $p+q$ is odd. Here, $H^{(p)}_{k/n}$ denotes the generalized harmonic number of order $p$ evaluated at the rational argument $k/n$.

This work is motivated by a known identity established by Sofo [11, Corollary 2.4] for the case $a=1$, which expresses such sums in terms of other Euler sums and products of Riemann zeta values. While the specific case $n=2$ had been previously evaluated in closed form, and the inner sums appearing in Sofo's identity had recently been resolved [7], a general closed-form expression for the broader class of sums with arbitrary rational arguments $k/n$ and alternating signs ($a=-1$) remained to be fully derived in a unified framework.

Methodology
The author employs a combination of partial fraction decomposition, series manipulation, and integral representations to derive the main results. The methodology proceeds through the following steps:

1. Partial Fraction Decomposition: Lemma 1 establishes a partial fraction identity for the term $\frac{1}{k^q(k+nj)^p}$. This allows the separation of the denominator into simpler components involving powers of $k$ and $(k+nj)$.
2. Series Transformation: The generalized harmonic number $H^{(p)}_{k/n}$ is expressed using its infinite series representation involving the Riemann zeta function: $H^{(p)}_{k/n} = \zeta(p) - \sum_{j=1}^{\infty} \frac{1}{(j+k/n)^p}$. Substituting this into the main sum transforms the problem into a double summation.
3. Application of Lemmas:
   • Lemma 2 provides an evaluation for sums of the form $\sum \frac{a^k}{(k+n)^p}$ in terms of the polylogarithm function $\text{Li}_p(a)$ and generalized harmonic numbers with alternating signs.
   • Lemma 3 (cited from [7, Corollary 1]) provides a closed-form evaluation for the specific Euler sums $\sum \frac{a^k H^{(p)}_{nk,b}}{k^q}$, which appear as intermediate terms in the derivation.
4. Interchange of Summation: By interchanging the order of summation and applying the lemmas, the original complex sum is decomposed into a linear combination of polylogarithms, the Lerch transcendent $\Phi$, and products of zeta functions.
5. Integral Derivation: Theorem 2 connects these series results to a related logarithmic integral involving the polylogarithm function, utilizing the integral representation of harmonic numbers.

Key Contributions and Results

• Theorem 1: The paper derives a closed-form expression for the general Euler sum $\sum_{k=1}^{\infty} \frac{a^k H^{(p)}_{k/n}}{k^q}$. The result is expressed in terms of:

  • The polylogarithm function $\text{Li}_s(z)$.
  • The Lerch transcendent $\Phi(z, s, t)$.
  • The Riemann zeta function $\zeta(s)$.
  • The Hurwitz zeta function (implicitly via the Lerch transcendent).
  • The formula involves finite sums over binomial coefficients and indices $r$ and $\ell$.

• Corollaries:

  • Corollary 1 ($a=1$): The derived formula reduces exactly to Sofo's identity [11], validating the new generalization.
  • Corollary 2 ($a=-1$): The paper provides explicit evaluations for alternating Euler sums. These results are further specialized for cases where $n$ is even and where $n$ is odd, introducing the Dirichlet eta function $\eta(s)$ and the Dirichlet lambda function $\lambda(s)$.

• Theorem 2: A closed-form evaluation is obtained for the logarithmic integral:
  $$\int_0^1 \frac{\ln^{p-1}(x) \text{Li}_q(ax)}{x(1-x^n)} dx$$
  This result is shown to be directly related to the Euler sums derived in Theorem 1.

• Explicit Examples: Section 4 provides concrete evaluations for specific integer parameters $(p, q, n)$. These examples demonstrate that the general formulas reduce to combinations of zeta values (e.g., $\zeta(3), \zeta(5), \zeta(7)$), products of zeta functions, and values of the Hurwitz zeta function with rational arguments (e.g., $\zeta(s, 1/3), \zeta(s, 1/5)$).

Significance and Claims
The paper claims to provide a unified framework for evaluating a broad class of Euler sums that were previously only partially understood or restricted to specific parameters ($n=1$ or $n=2$). By expressing these sums in terms of the polylogarithm and Lerch transcendent, the work demonstrates that they are reducible to standard special functions (Riemann and Hurwitz zeta functions) under admissible parameter choices.

The author notes that the derived formulas are suitable for computational implementation, providing specific Mathematica conventions (e.g., setting $\text{Li}_1(1)=0$ and $\text{Li}_0(1)=-1/2$) to handle singularities and ensure the formulas function correctly in symbolic computation software. The work does not propose new experimental applications but rather extends the theoretical landscape of Euler sums and their connections to polylogarithmic integrals.