Carlitz Multiple Dagger Polylogarithms

• Carlitz multiple dagger polylogarithms are multivariate function-field analogues of classical multiple polylogarithms defined over finite field arithmetic with unique summation conventions and sign factors.
• They are constructed using admissible indices and specialized power series that satisfy shuffle and stuffle relations, leading to nontrivial involutions on spaces of multiple zeta values.
• These polylogarithms underpin key advances in t-motivic transcendence, algebraic independence, and the duality phenomena in function-field multizeta theory.

Carlitz multiple “dagger” polylogarithms (CMDPL) constitute a distinguished class of multivariate, function-field analogues of classical multiple polylogarithms, defined over the arithmetic of positive characteristic. These objects are characterized by unique summation conventions and sign factors, resulting in a family of special values that play a central role in the structure theory of multiple zeta values (MZVs) attached to the Carlitz module. CMDPLs are key to the construction of nontrivial algebra involutions on spaces of function-field MZVs, reveal new dualities, and connect to deep questions in $t$-motivic transcendence and function-field arithmetic.

1. Algebraic Framework and Admissible Indices

Let $\mathbb F_q$ be a finite field of cardinality $q$, $A = \mathbb F_q[\theta]$ the associated polynomial ring, $k = \mathbb F_q(\theta)$ its function field, and $k_\infty = \mathbb F_q((\theta^{-1}))$ the completion at the infinite place. For $d \ge 1$, set

$L_d = \prod_{i=1}^d (\theta - \theta^{q^i}) \in A$

and define $A_+$ as the monic polynomials in $A$.

Given a multi-index $s = (s_1, ..., s_r)$, the sets

• $S_d(s) = \sum_{a \in A_+, \deg a = d} a^{-s}$,
• $L_d^{-s} = 1 / L_d^s$, appear as “building blocks” in polylogarithmic and zeta sums.

The admissible index set per Thakur is

$$I = \{ (s_1, ..., s_r) \mid r \ge 0,\, s_1, ..., s_{r-1} \leq q,\, s_r \leq q-1 \}$$

with the empty index $\emptyset$ of weight and depth zero. This framework supports the construction and linear-algebraic manipulation of CMDPL values and their involutions (Mishiba, 1 Jan 2026).

2. Definition and Notational Distinctions

For $s = (s_1, ..., s_r) \in I$, the Carlitz multiple dagger polylogarithm is defined by the formal power series

$$\mathrm{Li}^{\dagger}_s(z_1, ..., z_r) = (-1)^r\, \sum_{0 \leq d_1 \leq \cdots \leq d_r} \frac{z_1^{q^{d_1}}\cdots z_r^{q^{d_r}}} {L_{d_1}^{s_1} \cdots L_{d_r}^{s_r}} \in k[[z_1, ..., z_r]].$$

The essential features of the “dagger” notation are:

• Non-strict summation indices: $0 \leq d_1 \leq \cdots \leq d_r$,
• Global sign factor $(-1)^r$.

The special values at the “diagonal point” yield

$$\mathrm{Li}^{\dagger}_s(1) = (-1)^r\, \sum_{0 \leq d_1 \leq \cdots \leq d_r} \frac{1}{L_{d_1}^{s_1} \cdots L_{d_r}^{s_r}} \in k_\infty.$$

The associated multiple zeta dagger value is

$$\zeta_A^{\dagger}(s) = (-1)^r\, \sum_{0 \leq d_1 \leq \cdots \leq d_r} S_{d_1}(s_1) \cdots S_{d_r}(s_r) \in k_\infty.$$

For $s_i \leq q$ (all $i$), $\zeta_A^{\dagger}(s) = \mathrm{Li}^{\dagger}_s(1)$ (Mishiba, 1 Jan 2026).

3. Fundamental Algebraic Relations and Involution

CMDPLs, together with their special values, satisfy analogues of the classical shuffle and stuffle (harmonic product) relations:

• For all $P, Q \in h^1_R$ (the free concatenation algebra on indices),

$$\mathrm{Li}^{\dagger}_P(1)\, \mathrm{Li}^{\dagger}_Q(1) = \mathrm{Li}^{\dagger}_{P*Q}(1)$$

where $*$ denotes the stuffle product.

• In the case of depth $1$, the $q$-shuffle product due to Thakur governs the relations modulo $\zeta_A(q-1)$.

A central result is the existence of a nontrivial $R$-algebra involution

$$\iota: Z_R / \zeta_A(q-1)Z_R \to Z_R / \zeta_A(q-1)Z_R,$$

with $Z_R$ the $R$-span of all MZVs, such that

$$\iota(\mathrm{Li}_s(1)) = \mathrm{Li}^{\dagger}_s(1) \pmod{\zeta_A(q-1)} \quad \forall\, s.$$

This construction relies on the universal properties of the algebra $Z_R$, the concurrence of harmonic/stuffle relations and linear relations (arising from the $A^\bullet(s; m; n)$-relations) modulo $\zeta_A(q-1)$. The involution is involutive, i.e., $\iota^2 = \operatorname{id}$, and exchanges the non-dagger and dagger special values (Mishiba, 1 Jan 2026).

4. Examples and Explicit Computations

Several low-weight examples clarify the combinatorial and arithmetic nature of CMDPL:

• Weight 1, Depth 1:

$$\mathrm{Li}^{\dagger}_{(s_1)}(z_1) = -\mathrm{Li}_{(s_1)}(z_1).$$

At $z_1 = 1$,

$\zeta_A^{\dagger}(s_1) = -\zeta_A(s_1).$

The involution acts as negation modulo the period ideal $\zeta_A(q-1)$.

• Weight 2, Depth 2:

$$\mathrm{Li}^{\dagger}_{(s_1, s_2)}(z_1, z_2) = \sum_{0 \leq d_1 < d_2} \frac{z_1^{q^{d_1}} z_2^{q^{d_2}}}{L_{d_1}^{s_1} L_{d_2}^{s_2}} + \sum_{d \geq 0} \frac{z_1^{q^d} z_2^{q^d}}{L_d^{s_1+s_2}}.$$

At $(1, 1)$, this yields

$$\zeta_A^{\dagger}(s_1, s_2) = \sum_{d_1 < d_2} \frac{1}{L_{d_1}^{s_1} L_{d_2}^{s_2}} + \sum_{d \geq 0} \frac{1}{L_d^{s_1 + s_2}}.$$

• Fundamental Thakur Relation: The basic identity

$$\mathrm{Li}_q(1) - L_1\, \mathrm{Li}_{(1, q-1)}(1) = 0$$

is mirrored in the dagger family as

$$\mathrm{Li}^\dagger_q(1) - L_1\, \mathrm{Li}^\dagger_{(1, q-1)}(1) \equiv 0 \pmod{\zeta_A(q-1)}.$$

These examples demonstrate the involution’s effect and clarify the combinatorics specific to the “dagger” summation (Mishiba, 1 Jan 2026).

5. Structural Properties, Independence, and Basis Results

Key properties of the CMDPLs and their special values include:

• Both the set of non-dagger values $\{ \zeta_A(s)\mid s\in I\}$ and dagger values $\{ \zeta_A^\dagger(s)\mid s\in I\}$ form $R$-bases for $Z_R$ (as resolved by Chang–Chen–Mishiba and Im–Kim–Le–Ngo Dac–Pham).
• For each weight $w$,

$$\dim_R Z_{R,w} = \#\{ s\in I \mid \mathrm{wt}(s) = w \},$$

with analogous dimension formulas for the dagger space.

• Algebraic independence results: Papanikolas’ $t$-motive theory yields strong independence conclusions among Carlitz multiple polylogarithms and their dagger specializations. CMDPLs are thus essential for dimension, basis, and independence problems in positive characteristic multizeta theory (Mishiba, 1 Jan 2026).

6. Functional Equations and Generating Series

CMDPLs satisfy generating-series identities directly paralleling the classical duality for multiple polylogarithms and multizeta values. For each $s \in I$ of depth $r$,

$$\sum_{i=0}^{r} \mathrm{Li}_{s[:i]}(1)\, \zeta_A^\dagger(s[i+1:]) = \sum_{i=0}^{r} \zeta_A^\dagger(s[:i])\, \mathrm{Li}_{s[i+1:]}(1) = 0$$

where $s[:i]$ and $s[i+1:]$ denote the “left” and “right” truncations of the index. These identities govern the interplay between the two families of special values and are essential in understanding the full symmetry of the algebra of MZVs in positive characteristic (Mishiba, 1 Jan 2026).

7. Significance in Function-Field Arithmetic

CMDPLs naturally arise within the log-algebraicity of Anderson’s $t$-modules, the theory of function-field motives, and the study of $\infty$-adic and $v$-adic special values. They provide a mirror-image family of special values, linearly independent from the non-dagger values modulo $\zeta_A(q-1)$, and underpin the structure of the nontrivial involutive automorphism $\iota$. This involution realizes, for the first time in positive characteristic, an analogue of the unresolved classical problem of involutive automorphisms on the ring of (characteristic 0) multiple zeta values. CMDPLs are thus critical in advancing dimension, basis, and duality phenomena in function-field multizeta theory, closely linked to the $t$-motive paradigm and the theoretical framework of analytic continuation, monodromy, and functional relations developed for Carlitz multiple polylogarithms (Mishiba, 1 Jan 2026, Furusho, 2020).