Generalized Multiple Harmonic Sums

• Generalized multiple harmonic sums are iterated finite sums with variable weights that extend classical and alternating harmonic sums.
• They feature rich algebraic structures—including shuffle, stuffle, and duality relations—that connect polylogarithms and multiple zeta values.
• They enable analytic continuation, asymptotic expansions, and algorithmic evaluations, underpinning advances in quantum field theory and number theory.

A generalized multiple harmonic sum is an iterated finite sum of the form

$$S_{a_1,\ldots,a_k}(x_1,\ldots,x_k;n) = \sum_{n \geq i_1 \geq i_2 \geq \dots \geq i_k \geq 1} x_1^{i_1} i_1^{-a_1} \cdots x_k^{i_k} i_k^{-a_k}$$

where $a_j$ are positive integers (defining the "weight"), and $x_j \in \mathbb{K}^*$ for a field $\mathbb{K}$ of characteristic zero, generalizing the classical case $x_j = \pm 1$ (alternating harmonic sums). This construction extends to cyclotomic, binomially-weighted, $q$-analogs, and more general indexings, providing a rich algebraic and analytic structure that connects to polylogarithms, multiple zeta values, and deep number-theoretic congruences (Ablinger et al., 2013, Ablinger et al., 2013).

1. Definitions and Families of Generalized Multiple Harmonic Sums

The standard multiple harmonic sum of depth $r$ and weight $w = \sum s_i$ is

$$H_n(s_1, \ldots, s_r) = \sum_{n \geq i_1 > i_2 > \cdots > i_r \geq 1} \frac{1}{i_1^{s_1} \cdots i_r^{s_r}}$$

with $r=1$, $s_1=1$ reducing to the standard harmonic numbers. Including sign factors yields alternating/even "signed" sums (Rosen, 2013, Ablinger et al., 2013).

Generalized harmonic sums (S-sums) are defined as

$$S_{a_1,\ldots,a_k}(x_1,\ldots,x_k; n) = \sum_{n \geq i_1 \geq \dots \geq i_k \geq 1} x_1^{i_1} i_1^{-a_1} \cdots x_k^{i_k} i_k^{-a_k}$$

admitting arbitrary $x_j \in \mathbb{K}^*$. Specializations recover multiple polylogarithms, classical and alternating harmonic sums, and various other nested sums encountered in high-energy physics and combinatorics (Ablinger et al., 2013, Ablinger et al., 2013).

Cyclotomic and binomially-weighted sums are extensions where denominators become $m_j n+\ell_j$, with optional sign or root-of-unity weights, and numerators may include binomial coefficients:

$$S_{m,\ell,p}(N) = \sum_{n=1}^N \frac{ (\pm 1)^n }{ (m n + \ell)^p }$$

and similarly for nested/iterated forms (Ablinger et al., 2013).

$q$-analogs replace the denominator $k$ by the $q$-integer $[k]_q = (1-q^k)/(1-q)$ and introduce $q$-power weights, leading to significant arithmetic and combinatorial consequences in modular forms, supercongruence, and finite multiple zeta value theory (Zhao, 2013, Takeyama et al., 2020, Seki et al., 2018).

Generalized hyperharmonic numbers and "Euler sums" are higher-level generalizations obtained by recursively nesting classical harmonic numbers or by mixing zero and positive indices, often labeled as $S_{n}(\{0\}_r; t_1,\dots,t_k)$, and their summation over $n$ connects directly to multiple zeta values plus combinatorics (Stirling numbers of the first kind) (Xu, 2017, Xu, 2017).

2. Algebraic Structure: Shuffle, Stuffle, and Duality

Generalized multiple harmonic sums possess two interlocking algebraic structures:

• Stuffle (quasi-shuffle) algebra: For S-sums, the product expands as

$$S_{a}(x;n) S_{b}(y;n) = S_{a,b}(x,y;n) + S_{b,a}(y,x;n) + S_{a+b}(x y;n)$$

with recursive rules for nested depths, yielding a commutative and associative algebra generalizing that of polylogarithms (Ablinger et al., 2013, Ablinger et al., 2013).

• Shuffle algebra: On the side of iterated integrals or generalized polylogarithms (via Mellin/inverse Mellin), products decompose into shuffles of index words,

$$H_{a,\ldots}(x) H_{b,\ldots}(x) = \sum H_{\operatorname{shuffles}}(x)$$

providing fundamental relations among nested integrals that translate into identities for S-sums (Ablinger et al., 2013, Ablinger et al., 2013).

Dualities and Ohno-type relations: Deep symmetry identities relating sums with permuted or dual indices, exemplified by Hoffman's, Bradley's, and Ohno's dualities, hold for both finite $p$-truncated sums and their $q$-analogs. The Ohno-type identities provide powerful relations in the algebra of multiple zeta values and their finite versions, controlling the structure of congruences and "A-finite" MZVs (Seki et al., 2018, Takeyama et al., 2020).

3. Analytic Continuation, Mellin Transform, and Asymptotics

The Mellin (and inverse Mellin) transform links S-sums to generalized polylogarithms:

$$S_{a_1, \ldots, a_k}(x_1, \ldots, x_k; n) \longleftrightarrow \int_{0}^{1} x^n f(x) dx$$

where $f$ is formed by iterated integrals. This connection underpins asymptotic analysis, analytic continuation ($n \rightarrow \mathbb{C}$), and conversion between sum and integral representations (Ablinger et al., 2013, Ablinger et al., 2013).

Asymptotic expansions: Sums like $S_1(c;n) = \sum_{i=1}^n c^i / i$ admit full expansions for large $n$ by Euler-Maclaurin or Mellin analysis, facilitating applications in quantum field theory and transcendental number classification (Ablinger et al., 2013).

4. Arithmetic and Congruence Properties

Multiple congruence relations, often modulo powers of primes or cyclotomic polynomials in the $q$-case, govern the arithmetic of generalized multiple harmonic sums:

• Binomial coefficient congruences: For example,

$$\binom{kp-1}{p-1} \equiv \sum_{j=0}^{n} b_{j,n}(k)p^j H_{p-1}(\{1\}^j) \pmod{p^{2n+3}}$$

where the $b_{j,n}(k)$ are explicit polynomials, subsuming Wolstenholme's and Glaisher's results (Rosen, 2013, Levaillant, 2021).

• Hyperplane sums and Bernoulli polynomials: Characteristic congruences of the form

$\sum_{\substack{k_1+\cdots+k_n=p\k_i > 0}} \frac{a(k_1,...,k_n)}{k_1...k_n} \equiv C \cdot B_{p-n}\left(\frac{1}{3}\right) \pmod{p}$

tightly link harmonic sum evaluations on discrete simplices to Bernoulli polynomials (Ma et al., 2023).

• $q$-Wolstenholme and supercongruences: The $q$-analog setting enables congruences in $\mathbb{Z}[q]/([p]_q)$ which specialize to classical harmonic sum congruences as $q \to 1$ (Zhao, 2013, Takeyama et al., 2020).
• Finite and symmetric multiple zeta values: The formal limits of truncated multiple harmonic sums at prime roots of unity or at analytic cyclic roots realize Kaneko–Zagier's finite/symmetric MZV correspondence and encode deep congruences (Takeyama et al., 2020, Seki et al., 2018).

5. Relations to Multiple Zeta Values and Special Constants

Taking the $N \to \infty$ limit (with convergence) in generalized multiple harmonic sums produces special numbers, notably:

• Multiple zeta values (MZVs): Limits such as

$$\zeta_{a_1, ..., a_k} = \lim_{N\to\infty} S_{a_1, ..., a_k}(N)$$

are central in arithmetic geometry and quantum field theory, with their algebra controlled by shuffle and stuffle relations up to high weight (Ablinger et al., 2013, Hoffman, 2016).

• Cyclotomic and binomially-weighted extensions: Constants arising from sum limits at roots of unity ($\psi^{(k)}(r/s)$, polylogarithms, Clausen-type values, elliptic constants) generalize MZVs (Ablinger et al., 2013).
• Symmetric and quasi-symmetric function evaluations: Sums in harmonic numbers and their symmetric functions reduce to explicit combinations of MZVs, as in

$$\sum_{n=1}^{\infty} \frac{Q_\ell(H_n^{(1)},...,H_n^{(\ell)})}{n^2} = (\ell+1) \zeta(\ell+2)$$

for $Q_\ell$ the $\ell$th complete symmetric polynomial (Hoffman, 2016).

6. Algorithmic and Computational Aspects

Explicit analytic, algebraic, and recursive properties are encoded in computer algebra systems such as the Mathematica package HarmonicSums, providing:

• Symbolic manipulation of S-sums, Z-sums, polylogarithms, and cyclotomic variants.
• Automatic reduction to canonical bases via quasi-shuffle, shuffle, and duality relations.
• Mellin/inverse Mellin representation and analytic continuation for arbitrary complex $n$.
• Asymptotic expansions, basis computation, and identification of algebraic relations across sums and special values (Ablinger et al., 2013).

This computational infrastructure supports symbolic evaluation in high-loop quantum field theory, combinatorics, and arithmetic geometry.

7. Research Directions and Applications

Generalized multiple harmonic sums underpin deep results in:

• Modular forms, congruences in number theory, and Bernoulli/Bernoulli polynomial arithmetic.
• Multi-loop calculations in quantum field theory, via relation to harmonic polylogarithms and Mellin transforms (Ablinger et al., 2013, Ablinger et al., 2013).
• Finite and symmetric MZVs, supercongruence theory, and the structure of $q$-deformations of algebraic and arithmetic objects (Takeyama et al., 2020, Seki et al., 2018).
• Explicit identities and reductions in terms of symmetric functions, Stirling numbers, and combinatorics (Hoffman, 2016, Xu, 2017, Xu, 2017).
• Uniform congruence families and generalizations of classical arithmetical theorems (Wilson, Glaisher, Sun, Wolstenholme) to higher orders and to more elaborate twisted sums involving residue characters and polynomial weights (Ma et al., 2023, Levaillant, 2021).

Theoretical developments continue to elucidate the algebraic and analytic foundations, explicit reduction algorithms for broad classes of S-sums, and number-theoretic implications such as the construction and classification of bases of MZVs and their generalizations.