Parity Theorem for MPLs

• The parity theorem for multiple polylogarithms is a functional identity that reduces depth by expressing an MPL and its inverse in terms of lower-depth polylogarithms and Bernoulli-type terms.
• It is established through recursive analytic techniques, using differentiation and regularization, and is further validated by computational implementations.
• The theorem extends to coloured and motivic frameworks, linking special values, cyclotomic settings, and single-valued polylogarithmic functions in various applications.

Searching arXiv for recent and foundational papers on the parity theorem for multiple polylogarithms. The parity theorem for multiple polylogarithms concerns the depth-reducing functional equations satisfied by

$$\operatorname{Li}_{n_1,\ldots,n_d}(z_1,\ldots,z_d) =\sum_{0<k_1<\cdots<k_d}\frac{z_1^{k_1}\cdots z_d^{k_d}}{k_1^{n_1}\cdots k_d^{n_d}},$$

viewed either as analytic functions, special values at roots of unity, or motivic objects. In its basic form, the theorem states that a suitable parity combination of an MPL and the same MPL at inverted arguments can be written in terms of lower-depth polylogarithms, Bernoulli-type terms, and products; when $z_i=1$, this recovers the classical parity theorem for multiple zeta values (Panzer, 2015, Umezawa, 4 Aug 2025).

1. Definition of the parity phenomenon

For $\mathbf{k}=(k_1,\ldots,k_d)\in \mathbb{Z}_{>0}^d$ and $\mathbf{z}=(z_1,\ldots,z_d)\in\mathbb{C}^d$, the multiple polylogarithm is

$\Li_{\mathbf{k}}(\mathbf{z}) =\sum_{0<m_1<\cdots<m_d}\frac{z_1^{m_1}\cdots z_d^{m_d}}{m_1^{k_1}\cdots m_d^{k_d}},$

with absolute convergence if $|z_{i,d}|<1$ for all $i=1,\ldots,d$, where $z_{i,j}:=z_i\cdots z_j$ (Umezawa, 4 Aug 2025). Its weight is $|\mathbf{k}|=k_1+\cdots+k_d$, and its depth is $d$.

Panzer formulates the functional parity combination as

$z_i=1$0

where $z_i=1$1 (Panzer, 2015). The main theorem states that for all depths $z_i=1$2 and indices $z_i=1$3,

$z_i=1$4

where $z_i=1$5 denotes the $z_i=1$6-module spanned by products of Bernoulli polynomials in $z_i=1$7, powers of $z_i=1$8, and lower-depth MPLs, with total depth $z_i=1$9 and total weight $\mathbf{k}=(k_1,\ldots,k_d)\in \mathbb{Z}_{>0}^d$0 (Panzer, 2015).

This formulation shifts the classical parity theorem from a statement about special values to a statement about functional equations. In the MZV specialization $\mathbf{k}=(k_1,\ldots,k_d)\in \mathbb{Z}_{>0}^d$1, opposite parity of weight and depth forces a depth drop. In the general MPL setting, the same phenomenon is encoded by inversion $\mathbf{k}=(k_1,\ldots,k_d)\in \mathbb{Z}_{>0}^d$2 and the sign $\mathbf{k}=(k_1,\ldots,k_d)\in \mathbb{Z}_{>0}^d$3.

2. Analytic functional equations and recursive proof

The depth-one case is governed by Jonquière’s inversion formula,

$\mathbf{k}=(k_1,\ldots,k_d)\in \mathbb{Z}_{>0}^d$4

which already exhibits the parity pattern: the obstruction to exact antisymmetry is a Bernoulli polynomial in $\mathbf{k}=(k_1,\ldots,k_d)\in \mathbb{Z}_{>0}^d$5 (Panzer, 2015). The higher-depth theorem is proved recursively on the weight by combining differentiation formulas for MPLs, explicit primitives involving Bernoulli polynomials, and regularization arguments at singular points (Panzer, 2015).

Panzer’s result gives explicit formulas in depths $\mathbf{k}=(k_1,\ldots,k_d)\in \mathbb{Z}_{>0}^d$6 and $\mathbf{k}=(k_1,\ldots,k_d)\in \mathbb{Z}_{>0}^d$7, and the preprint emphasizes that the proof is constructive and implemented computationally: it provides a computer program to compute the functional equations (Panzer, 2015). The same work also states an additional integrality property for the MZV specialization, strengthening the classical parity theorem beyond mere existence of a depth reduction (Panzer, 2015).

In this analytic framework, the parity theorem is not simply a relation modulo products. It is an actual functional identity in the variables $\mathbf{k}=(k_1,\ldots,k_d)\in \mathbb{Z}_{>0}^d$8, with branch behavior controlled by Bernoulli polynomials and powers of $\mathbf{k}=(k_1,\ldots,k_d)\in \mathbb{Z}_{>0}^d$9. That viewpoint is essential for later extensions to coloured MZV, cyclotomic settings, and regularized values.

3. Explicit formulas and regularized extensions

A central later development is the transition from constructive recursion to a closed explicit formula. Umezawa gives an explicit parity formula for MPLs, inspired by Hirose’s explicit parity theorem for MZVs via multitangent functions (Umezawa, 4 Aug 2025, Hirose, 2024). Writing

$\mathbf{z}=(z_1,\ldots,z_d)\in\mathbb{C}^d$0

the formula takes the form

$\mathbf{z}=(z_1,\ldots,z_d)\in\mathbb{C}^d$1

with the paper giving the precise definitions of the star-sums and tilde-sums appearing on the right-hand side (Umezawa, 4 Aug 2025). In depth $\mathbf{z}=(z_1,\ldots,z_d)\in\mathbb{C}^d$2, this reduces to

$\mathbf{z}=(z_1,\ldots,z_d)\in\mathbb{C}^d$3

which is the classical inversion law in this normalization (Umezawa, 4 Aug 2025).

The same paper extends the formula to regularized values, both stuffle and shuffle, by introducing $\mathbf{z}=(z_1,\ldots,z_d)\in\mathbb{C}^d$4 and an explicit boundary term $\mathbf{z}=(z_1,\ldots,z_d)\in\mathbb{C}^d$5 to handle singular limits such as $\mathbf{z}=(z_1,\ldots,z_d)\in\mathbb{C}^d$6 (Umezawa, 4 Aug 2025). This is important because many arithmetic applications require precisely those limiting regimes.

Hirose’s MZV result is the corresponding explicit formula at $\mathbf{z}=(z_1,\ldots,z_d)\in\mathbb{C}^d$7. It states that if $\mathbf{z}=(z_1,\ldots,z_d)\in\mathbb{C}^d$8, then

$\mathbf{z}=(z_1,\ldots,z_d)\in\mathbb{C}^d$9

is given by an explicit sum whose coefficients involve Bernoulli numbers and powers of $\Li_{\mathbf{k}}(\mathbf{z}) =\sum_{0<m_1<\cdots<m_d}\frac{z_1^{m_1}\cdots z_d^{m_d}}{m_1^{k_1}\cdots m_d^{k_d}},$0, and every term on the right has depth strictly less than $\Li_{\mathbf{k}}(\mathbf{z}) =\sum_{0<m_1<\cdots<m_d}\frac{z_1^{m_1}\cdots z_d^{m_d}}{m_1^{k_1}\cdots m_d^{k_d}},$1 (Hirose, 2024). The MPL formula of Umezawa generalizes this from MZVs to arbitrary arguments.

4. Special values, roots of unity, and broader generalizations

Panzer’s theorem specializes to roots of unity, giving parity theorems for special values of MPLs at roots of unity, also known as coloured MZV (Panzer, 2015). The functional identity survives the specialization, after regularization when needed, and yields explicit depth reductions in the cyclotomic setting.

A contour-integral approach sharpens this direction. The paper on cyclotomic Euler sums and multiple polylogarithm functions states that

$\Li_{\mathbf{k}}(\mathbf{z}) =\sum_{0<m_1<\cdots<m_d}\frac{z_1^{m_1}\cdots z_d^{m_d}}{m_1^{k_1}\cdots m_d^{k_d}},$2

is always a combination of multiple polylogarithms of lower depth, for all $\Li_{\mathbf{k}}(\mathbf{z}) =\sum_{0<m_1<\cdots<m_d}\frac{z_1^{m_1}\cdots z_d^{m_d}}{m_1^{k_1}\cdots m_d^{k_d}},$3, and it supplies explicit formulas for low depths in the cyclotomic case (Rui et al., 30 Aug 2025). The method uses generalized digamma functions, extended trigonometric functions, and residue computations.

The parity phenomenon also sits inside a larger class of multiple Dirichlet-series identities. Kadota proves a parity result for a general family

$\Li_{\mathbf{k}}(\mathbf{z}) =\sum_{0<m_1<\cdots<m_d}\frac{z_1^{m_1}\cdots z_d^{m_d}}{m_1^{k_1}\cdots m_d^{k_d}},$4

with a functional equation expressing a depth-$\Li_{\mathbf{k}}(\mathbf{z}) =\sum_{0<m_1<\cdots<m_d}\frac{z_1^{m_1}\cdots z_d^{m_d}}{m_1^{k_1}\cdots m_d^{k_d}},$5 object in terms of lower-depth terms when the total weight has parity opposite to $\Li_{\mathbf{k}}(\mathbf{z}) =\sum_{0<m_1<\cdots<m_d}\frac{z_1^{m_1}\cdots z_d^{m_d}}{m_1^{k_1}\cdots m_d^{k_d}},$6 (Kadota, 2019). The paper describes this framework as encompassing the parity theorem for multiple polylogarithms because the exponential twists and generating-function coefficients can, under specialization and analytic continuation, be related to MPLs (Kadota, 2019).

These generalizations show that the parity theorem is not confined to the Euler–Zagier series. It is part of a broader pattern in which inversion, reflection, or sign-reversal symmetries force depth reduction across several families of period-like functions.

5. Motivic, coalgebraic, and fixed-weight formulations

In a motivic setting, parity becomes an inversion relation in a Lie coalgebra generated by symbols $\Li_{\mathbf{k}}(\mathbf{z}) =\sum_{0<m_1<\cdots<m_d}\frac{z_1^{m_1}\cdots z_d^{m_d}}{m_1^{k_1}\cdots m_d^{k_d}},$7 intended to model motivic MPLs (Greenberg et al., 2022). The cobracket is defined from Goncharov’s coproduct, and $\Li_{\mathbf{k}}(\mathbf{z}) =\sum_{0<m_1<\cdots<m_d}\frac{z_1^{m_1}\cdots z_d^{m_d}}{m_1^{k_1}\cdots m_d^{k_d}},$8 is proved for the full symbolic coalgebra (Greenberg et al., 2022).

At depth $\Li_{\mathbf{k}}(\mathbf{z}) =\sum_{0<m_1<\cdots<m_d}\frac{z_1^{m_1}\cdots z_d^{m_d}}{m_1^{k_1}\cdots m_d^{k_d}},$9, the parity relation becomes

$|z_{i,d}|<1$0

up to products (Greenberg et al., 2022). At depth $|z_{i,d}|<1$1, the corresponding relation is

$|z_{i,d}|<1$2

and Theorem 4.5 in that paper gives a general recursive expression for every inverted symbol in terms of non-inverted generators and lower-depth symbols (Greenberg et al., 2022). In that sense, all parity theorems for motivic MPLs become structural relations in the coalgebra.

Fixed-weight reduction problems provide another manifestation. In weight $|z_{i,d}|<1$3, Gangl shows that parity-like antisymmetry properties such as

$|z_{i,d}|<1$4

play a decisive role in reducing depth-$|z_{i,d}|<1$5 weight-$|z_{i,d}|<1$6 iterated integrals to linear combinations of $|z_{i,d}|<1$7-terms (Gangl, 2016). The paper proves Goncharov’s conjectural reduction of $|z_{i,d}|<1$8 to $|z_{i,d}|<1$9 tetralogarithm terms (Gangl, 2016).

In weight $i=1,\ldots,d$0, depth $i=1,\ldots,d$1, the parity theme appears as anharmonic symmetry. The function $i=1,\ldots,d$2 satisfies the six-fold transformations $i=1,\ldots,d$3 and $i=1,\ldots,d$4 in each variable independently, modulo terms of depth $i=1,\ldots,d$5 (Charlton, 2024). Together with earlier work on the “higher Gangl” part, this establishes Goncharov’s Depth Conjecture in the case of weight $i=1,\ldots,d$6, depth $i=1,\ldots,d$7 (Charlton, 2024). This suggests that the parity theorem is one facet of a broader symmetry-controlled depth-drop mechanism.

6. Single-valued, discrete, geometric, and physical contexts

Single-valued multiple polylogarithms provide a closely related parity theory. For the single-valued basis elements $i=1,\ldots,d$8, Schnetz records the congruence

$i=1,\ldots,d$9

and the associated MZV relation

$z_{i,j}:=z_i\cdots z_j$0

where $z_{i,j}:=z_i\cdots z_j$1 denotes reversal (Schnetz, 2013). These parity drops are used in the study of graphical functions and Feynman periods, including the proof of the zig-zag conjecture (Schnetz, 2013).

A discrete analogue survives before taking any limit to analytic MPLs. The discretization theorem

$z_{i,j}:=z_i\cdots z_j$2

identifies a finite multiple sum with a discrete iterated-integral expression, and the same work proves duality statements with parity signs $z_{i,j}:=z_i\cdots z_j$3 and $z_{i,j}:=z_i\cdots z_j$4 for MPLs and finite multiple zeta values (Hirose et al., 2024). This shows that parity is visible already at the level of discrete approximations.

Geometric period constructions also point in the same direction. For hyperplane arrangements with enough modular elements, associated periods are linear combinations of MPL values, often at roots of unity, and the paper remarks that arrangement symmetries can induce functional equations among those MPLs, including instances of the parity theorem (Tosi, 27 Feb 2026). This suggests a geometric source for some parity identities.

Across these settings, the parity theorem functions as a structural depth-reduction principle. Analytically, it is an inversion formula for MPLs; arithmetically, it controls MZV and coloured MZV reductions; motivically, it becomes a relation in a Lie coalgebra; geometrically and physically, it organizes special values arising from periods and Feynman amplitudes. The modern literature therefore treats parity not as an isolated identity but as a recurrent mechanism linking inversion symmetries, coproduct structures, and explicit depth-lowering formulas (Panzer, 2015, Umezawa, 4 Aug 2025).