Multiple Polylogarithm Functions

• Multiple polylogarithm functions are defined via nested series and iterated integrals, generalizing classical polylogarithms and linking to mixed Tate motives and multiple zeta values.
• They exhibit robust algebraic properties through shuffle and stuffle products, which provide graded Hopf algebra structures and facilitate explicit functional reductions.
• Their analytic framework, including convergence domains, meromorphic continuation, and regularization techniques, extends classical definitions to a broader range of applications.

Multiple polylogarithm functions (MPLs) are a fundamental class of special functions characterized by rich algebraic, analytic, and arithmetic structures. They arise in number theory, algebraic geometry, mathematical physics, and the computation of Feynman integrals. MPLs generalize the classical polylogarithm by introducing multiple indices and multiple complex variables, and are deeply tied to the structure of mixed Tate motives, iterated integrals, and the theory of multiple zeta values. Their algebraic and functional properties have far-reaching implications for research on periods, modular forms, and quantum field theory.

1. Definitions and Foundational Representations

The canonical definition of MPLs is via their nested series expansion: $\Li_{s_1,\dots,s_k}(z_1,\dots,z_k) = \sum_{n_1>\cdots>n_k\ge1} \frac{z_1^{n_1}\cdots z_k^{n_k}}{n_1^{s_1}\cdots n_k^{s_k}},$ where $s_1,\ldots,s_k$ are positive integers, $z_1,\ldots,z_k$ are complex variables with $|z_1\cdots z_i|<1$ for all $i$ for absolute convergence (Au, 2022, Frellesvig, 2018).

Equivalently, MPLs admit an iterated-integral representation: $\Li_{s_1,\dots,s_k}(z_1,\dots,z_k) = (-1)^k \int_0^1 \omega(0)^{s_1-1}\omega(a_1) \cdots \omega(0)^{s_k-1}\omega(a_k),$ where $\omega(a) = dx/(x-a)$, $a_i = (z_1\cdots z_i)^{-1}$ (Au, 2022, Kaneko et al., 2024). This exhibits the role of Chen’s theory of iterated integrals and connects MPLs with the algebra of paths and fundamental groups of $\mathbb{P}^1$ minus points.

MPLs extend classical special cases: $\Li_n(z)$ (classical polylogarithm) for $k=1$, and multiple zeta values when all variables are $1$: $\zeta(s_1,\dots,s_k) = \Li_{s_1,\dots,s_k}(1,\ldots,1)$ for $s_k \geq 2$.

2. Algebraic Structures and Functional Equations

MPLs exhibit two central algebraic structures:

• Shuffle Product: Derived from the iterated-integral representation, leading to

$$G(u;z)\,G(v;z) = \sum_{\sigma\in\mathrm{Sh}(r,s)} G(\sigma(a_1,\dots,a_r,b_1,\dots,b_s);z),$$

yielding linear relations (shuffle relations) among MPLs (Frellesvig, 2018, Bogner et al., 2012).

• Stuffle (Quasi-shuffle) Product: From the series definition, the product of MPLs expands canonically into linear combinations of MPLs with varying depths but fixed total weight

$\Li_m(x)\,\Li_n(y) = \Li_{m,n}(x,y) + \Li_{n,m}(y,x) + \Li_{m+n}(xy)$

and more generally via combinatorial stuffle rules (Frellesvig, 2018).

Both shuffle and stuffle structures equip the set of MPLs with a graded Hopf algebra structure, allowing for deep algebraic manipulations, symbol calculus, and explicit functional reduction algorithms. Important relations include inversion formulas (e.g., for the dilogarithm), dualities, and connections to Bernoulli polynomials and Stieltjes constants. In particular, the generalized parity theorem expresses

$\Li_{s_1,\dots,s_d}(z_1,\dots,z_d) - (-1)^{w-d} \Li_{s_1,\dots,s_d}(1/z_1,\dots,1/z_d)$

as a sum of products of lower-depth MPLs and logarithms (Panzer, 2015, Rui et al., 30 Aug 2025).

Recent work establishes combinatorial-functional equations of polygonal type that allow for explicit depth-reduction at fixed weight, with implications for the depth filtration in the motivic Lie algebra generated by MPLs (Charlton et al., 2020).

3. Analytic Aspects, Domains of Convergence, and Regularization

The domain of absolute convergence of the nested series for $\Li_{(z_1, \ldots, z_r)}(s_1, ..., s_r)$ is given by $\mathrm{Re}(s_1+...+s_i)>i$ for all $i$ and $|z_j\cdots z_i|<1$ (Mehta et al., 2 Nov 2025, Mehta et al., 24 Jan 2026). However, significant progress has been made in extending the definition:

• Conditional Convergence Domains: By precise translation formulas, convergence is extended to a larger open set $U_r(z_1, ..., z_r)$, where for $i\geq q(z)$, $\mathrm{Re}(s_1+...+s_i)>i-1$ with $q(z)$ the first index where $z_1\cdots z_i\neq 1$.
• Meromorphic Continuation and Singularities: MPLs extend meromorphically to $\mathbb{C}^r$ with poles on explicit hyperplanes determined by the products of $z$’s equaling 1 and integer values of the sum $s_1+\cdots+s_i$ (Mehta et al., 2 Nov 2025, Mehta et al., 24 Jan 2026).
• Regularization: Via a generalization of Euler-Boole summation, a regularization procedure defines consistent values and local Laurent expansions at integer points, extending the notion of Stieltjes constants and capturing the fine structure of singularities (Mehta et al., 24 Jan 2026, Mehta et al., 2 Nov 2025).

This analytic machinery allows for the treatment of special values, extends the classical polylogarithm theory to broader parameter domains, and supports the investigation of cyclotomic and alternating cases.

4. Colored Multiple Zeta Values and Algebraic Arguments

Evaluation of MPLs at algebraic arguments, particularly roots of unity, is of foundational significance. Suppose $a_i \in \{0,1,\mu,\ldots,\mu^{N-1}\}$, $\mu = e^{2\pi i/N}$, then

$\Li_{s_1,\dots,s_k}(a_1,\dots,a_k) \in \mathbb{Q}\text{-span of iterated integrals over } \omega(b_i), \; b_i\in S,$

defining the colored multiple zeta value (CMZV) algebra of level $N$ and weight $w$ (Au, 2022). For more general algebraic arguments, Möbius transformation and iterated-integral pullback provide systematic reduction to CMZVs.

The introduction of $N$-unital rational maps generates previously inaccessible non-standard linear relations among CMZVs, deepening the connection to Deligne’s dimension conjectures and revealing new algebraic structures in the graded $\mathbb{Q}$-algebra of CMZVs (Au, 2022).

Applications include:

• Uniform proofs of classical dilogarithm “ladders.”
• Evaluation of Apéry-type binomial sums, relating series such as $\sum c^n/n^s A_n$ to integrals of MPLs and explicitly to CMZVs, e.g.,

$$\sum_{n\ge1}\frac{(-1)^n}{n^3\binom{2n}{n}} = -\frac{2}{5}\zeta(3)$$

(Au, 2022).

5. Extensions: Higher-Variable, Elliptic, and Finite Polylogarithms

Two-Variable and Level-$N$ Cases

Generalization to two or more variables yields a variety of new function classes, such as Kaneko-Tsumura’s

$$L_{k_1,\dots,k_r}(x,y)= \sum_{0<m_1<\cdots<m_r} \frac{x^{m_r} (1-y^{m_1})\cdots(1-y^{m_r-m_{r-1}})} {m_1^{k_1}\cdots m_r^{k_r}}$$

interpolating classical, level-$N$, and other special cases. Weighted sum and connection-type identities generalize classical formulas and recover foundational identities (e.g., the five-term dilogarithm relation) as special cases (Kaneko et al., 2024).

Elliptic Multiple Polylogarithms

Elliptic generalizations are constructed by integrating classical multiple polylogarithm kernels over cycles on elliptic curves, e.g.,

$$I_\Gamma(\tau;\zeta) = \int_{E^{N-1}} \prod_{\text{edges}} D_{1,1}(\tau;\zeta_i/\zeta_j)$$

where $D_{1,1}$ is the Green function, and $I_\Gamma$ is a single-valued elliptic multiple polylogarithm. Such functions capture the analytic content of modular graph functions arising in the low-energy expansion of superstring amplitudes (D'Hoker et al., 2015). Depth/breadth reduction and Laurent expansions at the cusp are controlled by classical multiple polylogarithms and single-valued MZVs.

Remiddi–Tancredi developed elliptic MPLs as natural iterated integrals on genus one curves with explicit differential reduction and reduction to classical MPLs in the degenerate (nodal) limit (Remiddi et al., 2017).

Finite Multiple Polylogarithms

In the context of arithmetic and mod $p$ phenomena, finite multiple polylogarithms have been developed, extending the shuffle/stuffle structures and “$t\leftrightarrow 1-t$” functional equations to the finite field setting, with correction (error) terms reflecting the obstruction posed by positive characteristic (Ono, 2017).

6. Generating Functions, State-Space Realizations, and Algorithmic Computation

Generating functions for periodic MPLs can be encoded as rational Chen–Fliess series, allowing translation into finite-dimensional bilinear dynamical systems. For an admissible multi-index $s$, the generating function

$\mathcal{L}_s(t,\theta) = \sum_{n=0}^\infty \Li_{\{s\}^n}(t)\, [\theta^{|s|}]^n$

can be realized via a canonical companion-matrix construction. This underpins both highly efficient numerical algorithms (notably capable of verifying conjectures such as the Hoffman basis conjecture) and algebraic interpretations in dendriform algebra, which organizes the iterated integral shuffle structures (Ebrahimi-Fard et al., 2016).

Major symbolic computation platforms (e.g., Maple 2018 GeneralizedPolylog/ MultiPolylog) have implemented comprehensive suites for algebraic reduction (shuffle, stuffle, duality), special-value recognition, and high-precision numerical evaluation, leveraging these formalisms (Frellesvig, 2018).

7. Open Problems and Structural Insights

• Depth-Reduction and Motivic Structures: Polygonal functional equations in weights 5,6,7 enable explicit depth reduction for high-weight MPLs and connect to conjectures of Zagier and Goncharov regarding motivic polylogarithms (Charlton et al., 2020).
• Steinberg Module and Depth Filtration: Multiple polylogarithms on a torus are shown to be expressible as $\Li_{n-d+1,1,\dots,1}(x_1,\dots,x_d)$ up to product terms, linking MPLs to the top homology of Tits buildings (Steinberg module), and giving representation-theoretic proofs of classical theorems in $K$-theory (Charlton et al., 4 May 2025).
• Local Analytic Expansion and Regularization: In-depth analytic study at integer points enables a complete Laurent-type expansion, with coefficients given by regularized values of MPLs and explicit combinatorial factors, parallel to the theory of Stieltjes constants for the Riemann zeta function (Mehta et al., 24 Jan 2026).
• Parity and Cyclotomic Reductions: Recent contour integration methods yield parity reduction theorems for cyclotomic and classical MPLs, isolating lower-depth components and enabling explicit evaluations of alternating and colored zeta and polylogarithm values (Rui et al., 30 Aug 2025, Panzer, 2015).

Collectively, these developments have established multiple polylogarithm functions as a structured universe with deep interconnections spanning algebra, analysis, arithmetic, and mathematical physics, and provided effective computational and theoretical frameworks for current and future research.