Landen Formula for Complex Multiple Polylogarithms

• The paper generalizes classical Landen reflection by extending dilogarithm identities to arbitrary depth using a novel algebraic-geometric framework.
• It employs the symmetry of the punctured Riemann sphere and the Knizhnik–Zamolodchikov equation, leveraging non-analytic methods and combinatorial word techniques.
• The results reveal deep connections with S3-symmetry and support both complex and ℓ-adic analyses, advancing the study of multiple polylogarithm functional relations.

The Landen formula for complex multiple polylogarithms generalizes the classical Landen reflection for the dilogarithm to arbitrary depth, providing explicit functional equations relating values of multiple polylogarithms at $z$ and at the transformed argument $\frac{z}{z-1}$. These results are established on the Riemann sphere punctured at $\{0,1,\infty\}$, making essential use of the symmetry $z \mapsto \frac{z}{z-1}$, the formal Knizhnik–Zamolodchikov (KZ) equation, and the algebraic structure of their groupoid-valued fundamental solutions. The contemporary proof, as given by Shiraishi, eschews analytic methods in favor of algebraic and geometric techniques, yielding a formula valid for all multi-indices and compatible with both complex and $\ell$-adic realizations of multiple polylogarithms (Shiraishi, 16 Jan 2026). This formalism also integrates and clarifies connections with the $S_3$-symmetry and related reflection/duality equations (Shiraishi, 2023).

1. Preliminaries and Definitions

Let $d \geq 1$ and $\mathbf{k} = (k_1, \dots, k_d) \in \mathbb{N}^d$ denote a multi-index with weight $\mathrm{wt}(\mathbf{k}) = k_1 + \cdots + k_d$ and depth $\mathrm{dp}(\mathbf{k}) = d$. The refinement partial order $\mathbf{J} \preceq \mathbf{k}$ encodes the grouping of sub-indices whose sums recover $\mathbf{k}$'s components.

Multiple polylogarithms $Li_{k_1,\dots,k_d}(z;\gamma)$ are defined via iterated integrals along a path $\gamma$ in $\mathbb{P}^1(\mathbb{C}) \setminus \{0,1,\infty\}$ from a tangential base point $\overrightarrow{01}$ at $z=0$ to $z$, with the formalism implemented using non-commuting variables $e_0^{\mathrm{dR}}, e_1^{\mathrm{dR}}$ satisfying $$e_\infty^{\mathrm{dR}} = -e_0^{\mathrm{dR}} - e_1^{\mathrm{dR}}$$. The fundamental solution to the KZ equation,

$$\frac{d}{dz}G(z) = \left( \frac{e_0^{\mathrm{dR}}}{z} + \frac{e_1^{\mathrm{dR}}}{z-1} \right) G(z),$$

expands as a formal series in words, with the coefficients extracting the multiple polylogarithms through the mapping of specific word monomials $W_{\mathbf{k}}$ to $Li_{\mathbf{k}}(z;\gamma)$.

2. The Landen Formula for Complex Multiple Polylogarithms

Given a path $\gamma$ from $\overrightarrow{01}$ to $z$, define $\gamma' = \delta \cdot \phi(\gamma)$, where $\phi(z) = \frac{z}{z-1}$ and $\delta$ is a fixed path from $\overrightarrow{01}$ to the tangential base point at $\infty$ along the positive real axis. Then, for every multi-index $\mathbf{k}$,

$$Li_{\mathbf{k}}\left(\frac{z}{z-1};\,\gamma'\right) = (-1)^{\mathrm{dp}(\mathbf{k})} \sum_{\mathbf{J} \preceq \mathbf{k}} Li_{\mathbf{J}}(z;\gamma).$$

This statement generalizes the classical Landen formula for the dilogarithm to arbitrary depths and indices (Shiraishi, 16 Jan 2026).

3. Algebraic–Geometric Proof Structure

The derivation is grounded in the symmetry of $\mathbb{P}^1 \setminus \{0,1,\infty\}$ under $z \mapsto \frac{z}{z-1}$ and a groupoid-chain rule for the fundamental KZ solutions. Specifically:

• The path groupoid composition reflects the transformation of the argument and base points.
• The chain rule for KZ solutions (Lemma 3.3) gives

$$G^{\,\frac{z}{z-1},\,\gamma'}(e_0,e_1) = G^{\,z,\gamma}(e_0, -e_0-e_1)\,\exp(\pi i\,e_0).$$

• Expansion in the non-commutative word basis, followed by a combinatorial lemma on word substitution, allows extraction of coefficients corresponding to multiple polylogarithms at transformed arguments.
• The sign $(-1)^{\mathrm{dp}(\mathbf{k})}$ and the sum over refinements encode the combinatorial impact of the transformation on the polylogarithmic indices.

No analytic continuation or explicit manipulation of series expansions is needed; the entire argument is based on algebraic properties, shuffle relations, and the formal group-like structure of the KZ solutions (Shiraishi, 16 Jan 2026).

4. Explicit Examples in Low Weights

Weight 1 ($\mathbf{k} = (1)$)

The formula specializes to

$$Li_1\left(\frac{z}{z-1};\gamma'\right) = -Li_1(z;\gamma).$$

With $Li_1(z;\gamma) = -\ln(1-z)$, this recovers the classical relation

$\ln\left(1-\frac{z}{z-1}\right) = \ln(1-z).$

Weight 2 ($\mathbf{k} = (2)$)

Refinements are $(2)$ and $(1,1)$, yielding

$$Li_2\left(\frac{z}{z-1};\gamma'\right) + Li_2(z;\gamma) + Li_{1,1}(z;\gamma) = 0.$$

Since $Li_{1,1}(z) = \frac{1}{2} \ln^2(1-z)$, this reproduces the classical Landen formula for the dilogarithm,

$$Li_2\left(\frac{z}{z-1}\right) + Li_2(z) + \frac{1}{2} \ln^2(1-z) = 0.$$

These identities confirm the general formula in classical cases and demonstrate its compatibility with functional equations known for specific polylogarithmic weights (Shiraishi, 16 Jan 2026).

5. Comparison with $S_3$ Symmetries and Reflection Formulas

The Landen formula for multiple polylogarithms is a distinguished case of a broader class of $S_3$-symmetry-induced functional equations on $\mathbb{P}^1 \setminus \{0,1,\infty\}$ (Shiraishi, 2023). Generalizations include:

• Duality and reflection formulas under $z \mapsto 1-z$, $z \mapsto 1/z$, and $z \mapsto \frac{z}{z-1}$.
• These are proved algebraically using associator chain-rule identities for the KZ equation and Drinfeld associators.
• The coefficients in formal expansions obey shuffle product relations, giving rise to intricate relations among polylogarithmic values at $z$ and its $S_3$-related images.

Examples such as the trilogarithmic and higher-weight cases demonstrate the broader patterns and coherence of these reflection and duality formulas. The Landen formula emerges as the specific $z \mapsto \frac{z}{z-1}$ instance within this web of functional equations.

6. Extensions to $\ell$-adic Galois Analogues

Replacing the de Rham KZ solutions with pro-$\ell$ Galois 1-cocycles $$\mathfrak{f}^{z,\gamma} \in Z^1(G_K, \mathbb{Q}_\ell \langle\langle e_0, e_1 \rangle\rangle)$$, the Landen formula admits an $\ell$-adic analogue. In this setting, the formula acquires explicit lower-weight error terms resulting from higher-order terms in the Baker–Campbell–Hausdorff series $\log(e^{-e_1} e^{-e_0})$. These error terms are expressed via integrals involving Goldberg polynomials and reflect the subtler structure of the $\ell$-adic polylogarithmic Galois action (Shiraishi, 16 Jan 2026).

7. Historical Context and Significance

The analytic Landen formula for the dilogarithm and certain higher-weight cases were classically derived by Okuda and Ueno via power-series manipulations. The algebraic-geometric proof by Shiraishi and contemporaries replaces analytic continuation with formal groupoid and combinatorial methods, extending the result to arbitrary depths and elucidating the underlying topological symmetries. This perspective aligns the theory of multiple polylogarithms with the modern language of iterated integrals, fundamental groupoids, the KZ equation, and the Grothendieck–Teichmüller group, thereby integrating reflection and duality phenomena within a unified algebraic and Galois-theoretic framework (Shiraishi, 16 Jan 2026, Shiraishi, 2023).