Cyclotomic Multiple Hurwitz Polylogarithm Functions

• Cyclotomic multiple Hurwitz polylogarithm functions are a generalization of classical polylogs defined through nested series and iterated integrals with cyclotomic roots and Hurwitz shifts.
• They exhibit rich algebraic structures including shuffle, stuffle, and distribution relations, which enable symmetry reductions and analytic continuation.
• These functions have practical applications in arithmetic geometry, quantum field theory, and computational mathematics, aiding evaluations of special constants and Feynman integrals.

Cyclotomic multiple Hurwitz polylogarithm functions generalize classical multiple polylogarithms by introducing cyclotomic, multiple, and Hurwitz-shift features in their summation and integral representations. These functions, central to current research in arithmetic geometry, special function theory, and quantum field theory computations, exhibit rich algebraic structures, explicit analytic continuation, and parity-symmetry phenomena. Their connections to cyclotomic harmonic sums, colored multiple zeta values, and special values underpin advances in transcendental number theory and computational mathematics.

1. Foundational Definitions and Notation

Cyclotomic multiple Hurwitz polylogarithm functions are characterized by their series and iterated-integral formulations. For $r \in \mathbb{N}$ (depth), multi-weight ${\bf k} = (k_1,\dots, k_r)\in\mathbb{N}^r$ (total weight $w = \sum k_i$), and cyclotomic parameters ${\bf x}=(x_1,\dots,x_r)$ with each $x_i$ an $N$th root of unity (cyclotomic order), together with a Hurwitz shift $a \in \mathbb{C}\setminus\{0,-1,-2,\dots\}$, the core definition is: $\Li_{k_1,\dots,k_r}(x_1,\dots,x_r\,;\,a+1) = \sum_{0<n_1<\cdots<n_r} \frac{x_1^{n_1}\cdots x_r^{n_r}}{(n_1+a)^{k_1}\cdots(n_r+a)^{k_r}}$ This sum converges absolutely for all $|x_i|=1$ and $a \not\in \{0,-1,-2,\dots\}$.

Cyclotomic harmonic polylogarithms $C_{k_1,\dots,k_m}^{l_1,\dots,l_m}(x)$ are defined via iterated Poincaré integrals over the alphabet $f_k^l(x) = x^l/\Phi_k(x)$, where $\Phi_k(x)$ denotes the $k$th cyclotomic polynomial and $0\leq l < \varphi(k)$ with Euler's totient function $\varphi$. Hurwitz-shifted kernels may be introduced as $$f_{(k, l, a)}(x) = x^l / \Phi_k(xe^{-2\pi ia}) e^{-2\pi ia}$$ (Rui, 30 Dec 2025, Ablinger et al., 2011, Ablinger et al., 2017).

2. Series, Integral, and Algebraic Structures

Series and Integral Representations

The nested series expansion for these functions employs both root-of-unity weights and Hurwitz shifts: $\Li^{(N)}_{k_1,\dots,k_r}(z; a_1, \dots, a_r) = \sum_{n_1 > n_2 > \dots > n_r > 0} \frac{z^{n_1} \prod_{j=1}^{r-1} \omega_N^{n_j - n_{j+1}}} {(n_1 + a_1)^{k_1} \cdots (n_r + a_r)^{k_r}}$ Integral representations involve repeated Beta integrations and changes of variables, such as

$$(n + a)^{-s} = \frac{1}{\Gamma(s)} \int_0^\infty t^{s-1} e^{-(n+a)t} dt$$

yielding multidimensional ordered integrals after reparameterization (Ablinger et al., 2017).

Algebraic Relations: Shuffle, Stuffle, Distribution, Duplication

Cyclotomic multiple Hurwitz polylogarithms form a shuffle algebra under the concatenation of iterated integral words: $H(\mathbf{A}; x) H(\mathbf{B}; x) = \sum_{C \in \mathbf{A} \shuffle \mathbf{B}} H(C; x)$ Similarly, products of the nested sums obey the stuffle (quasi-shuffle) algebra, and distribution relations generalize standard polylogarithm identities to root-of-unity and shifted arguments. Duplication and reflection functional equations relate function values under $a \mapsto 1-a$, $z \mapsto 1-z$ and enable symmetry reductions (Rui, 30 Dec 2025, Ablinger et al., 2011, Ablinger et al., 2017).

3. Parity, Symmetry, and Contour Integration Techniques

The parity phenomena for cyclotomic multiple Hurwitz polylogarithms, as established in (Rui, 30 Dec 2025), are consequences of contour integration and residue calculus. Considering parametric Euler sums such as

$$S^{(a)}_{p_1, \dots, p_r; q}(x_1,\dots,x_r; x) = \sum_{n=1}^\infty \frac{\prod_{j=1}^r \zeta_n(p_j; x_j; a)}{(n+a)^q} x^n$$

where $\zeta_n(p;x;a)=\sum_{k=1}^n x^k/(k+a)^p$, telescoping and partial fraction decompositions map these sums explicitly to combinations of cyclotomic multiple Hurwitz polylogarithms.

Contour integrals of the form

$$\oint_{|s|=R \to \infty} \Phi(s; x) \prod_{j=1}^r \phi^{(p_j-1)}(s+a; x_j) \frac{(-1)^{\sum p_j - r}}{(s+a)^q} ds = 0$$

(where $\Phi$ and $\phi$ are digamma-type kernels) vanish by Jordan's lemma and residue computation at poles, deriving explicit parity reductions. For every depth $r$, signed sums involving $a$ and $-a$ shifts collapse to linear combinations of lower-depth terms.

Conjectures regarding parity and antipode symmetry mod products (Conjectures 6.1 and 6.2, (Rui, 30 Dec 2025)) extend these identities to arbitrary depth and cyclotomic order.

4. Analytic Continuation and Special Value Evaluations

Analytic continuation in $N$ is implemented via recurrences for Mellin kernels $\phi_k(l,N)$: $$\sum_{n=0}^{N_k} c_{n,k}\, \phi_k(l, N+n-l) = \frac{1}{N+1}$$ alongside factorial series expansions, leading to meromorphic continuation with poles at non-positive integers (Ablinger et al., 2011). Explicit asymptotic expansions, as for

$$\phi_1(0,N) \sim \ln N + \gamma + \frac{1}{2N} - \frac{1}{12N^2} + \dots$$

are given for polylogarithms and their cyclotomic analogues.

Special value computations at $x=1$ and for infinite sums connect these functions to colored multiple zeta values, Clausen integrals, and special logarithmic or digamma constants, as in the identification

$\Li_2(e^{2\pi i/3}) = \frac{\pi^2}{6} - \frac{\ln^2 3}{2} + i \Cl_2\left(\frac{\pi}{3}\right)$

and similar for higher cyclotomies (Ablinger et al., 2011, Ablinger et al., 2017). PSLQ and related symbolic searches verify completeness of such constant bases up to moderate weight (Ablinger et al., 2017).

5. Applications, Symmetric Identities, and Computational Techniques

Cyclotomic multiple Hurwitz polylogarithms appear in:

• Evaluation of Feynman integrals for quantum field theory calculations, requiring extensions of classical harmonic sums to general cyclotomic orders (Ablinger et al., 2011).
• Analysis of Apéry-like series, with contour integration expressing bilateral and one-sided hypergeometric series in terms of these functions and providing new functional identities for polylogarithms of algebraic arguments (Xu, 19 Dec 2025).
• Explicit combinatorial computation of special constants and basis reductions for weights and cyclotomies up to $l=20$ (Ablinger et al., 2011, Ablinger et al., 2017).

Techniques for high-precision numerical evaluation involve Maclaurin series, variable transformations (e.g., $x = \frac{1-t}{1+t}$), Bernoulli speed-up, and contour deformation for shifted cyclotomic kernels. Symbolic computation is facilitated by implementations in the HarmonicSums package (Ablinger et al., 2011).

6. Functional Equations, Conjectures, and Further Directions

Functional equations derive from product differentiation, classical inversion, duplication and reflection identities. Deep connections to Bernoulli polynomials, symbol-maps, Galois coaction, and motivic structures are conjectured to underlie the analytical properties and interrelationships of the cyclotomic multiple Hurwitz polylogarithms (Ablinger et al., 2017).

Two open problems concern the existence of closed forms for one-sided Apéry-type and parametric central binomial series in terms of special functions beyond current explicit results (Xu, 19 Dec 2025). Further work in these areas is expected to illuminate the symbolic and transcendental aspects of cyclotomic multiple Hurwitz polylogarithms, as well as their applications in advanced mathematical physics and number theory.

7. Summary Table: Algebraic and Analytic Properties

Property	Cyclotomic Multiple Hurwitz Polylogarithms	Classical Polylogarithms
Shuffle Algebra	Yes	Yes
Stuffle/Quasi-shuffle Algebra	Yes	Yes
Distribution Relations	Yes (roots of unity and shifts)	Yes (roots of unity, no shifts)
Parity Symmetry	Explicit (via contour residue)	Classical parity for MZVs
Analytic Continuation	Meromorphic (via Mellin/recurrence)	Standard for Hurwitz/Lerch zeta
Special Value Bases	Clausen, log, digamma, zeta, PSLQ-closed	Clausen, log, zeta

Cyclotomic multiple Hurwitz polylogarithm functions synthesize and generalize key aspects of polylogarithmic structures, providing powerful tools for explicit computation, symmetry analysis, and transcendental constant evaluation in advanced mathematical domains.