Twisted Multiple Bernoulli Numbers (TMBNs)

• TMBNs are multivariable, root-of-unity–twisted generalizations of Bernoulli numbers defined via exponential generating functions and realized as special values of generalized zeta-functions.
• They bridge complex multiple zeta theory, cyclotomic Bernoulli numbers, and p-adic multiple L-functions through explicit combinatorial identities and analytic continuations.
• Their properties include entire continuation in twisted cases, desingularization for untwisted limits, and parity-driven vanishing phenomena that extend classical results.

Searching arXiv for the cited papers and closely related work on twisted multiple Bernoulli numbers. Twisted multiple Bernoulli numbers (TMBNs) are multivariable, root-of-unity–twisted generalizations of Bernoulli numbers defined by exponential generating series and characterized equivalently as special values at non-positive integers of generalized Euler–Zagier–Lerch multiple zeta-functions. In the modern formulation, they mediate between complex multiple-zeta theory, cyclotomic multiple Bernoulli numbers, and the special values of $p$-adic multiple $L$-functions; recent work makes this mediation explicit by rewriting coefficients in Furusho–Jarossay expansions of $p$-adic multiple $L$-values in terms of TMBNs, or equivalently in terms of complex zeta-special values at non-positive integers (Fan, 5 Jul 2026). Their foundational analytic and arithmetic properties were developed systematically by Furusho, Komori, Matsumoto, and Tsumura, including analytic continuation, desingularization, congruences, vanishing phenomena, and links to $p$-adic twisted multiple polylogarithms (Furusho et al., 2013).

1. Definition by generating functions

For a root of unity $\xi$, the twisted single Bernoulli numbers are defined by

$$H(t;\xi)=\frac{1}{1-\xi e^t} =\sum_{n=-1}^{\infty} B_n(\xi)\frac{t^n}{n!}, \qquad (-1)!\equiv 1.$$

In particular,

$B_{-1}(\xi)=-1,$

and when $\xi=1$ one recovers the classical Bernoulli numbers through

$B_n(1)=-\frac{B_{n+1}}{n+1}\qquad (n\ge 0).$

In depth $L$0, with

$L$1

the generating function is

$L$2

The coefficient

$L$3

is the depth-$L$4 twisted multiple Bernoulli number.

An earlier notation writes the same family via

$L$5

so that the coefficient

$L$6

is the TMBN of index $L$7, twists $L$8, and scales $L$9 (Furusho et al., 2013).

These definitions already exhibit two characteristic features. First, the twisting data are cyclotomic. Second, the variables $p$0 enter through nested sums $p$1, so the depth-$p$2 theory is not a simple product of depth-one pieces, even though explicit low-depth formulas later recover such pieces combinatorially.

2. Special values of generalized Euler–Zagier–Lerch zeta-functions

The analytic counterpart of TMBNs is the generalized Euler–Zagier–Lerch multiple zeta-function

$p$3

which converges in the region

$p$4

If $p$5 for all $p$6, this function admits analytic continuation to all of $p$7, and indeed is entire. At tuples of non-positive integers one has the exact special-value formula

$p$8

or, in the notation of the more paper,

$p$9

This identity is the conceptual core of the subject. TMBNs are not merely coefficients extracted from a formal multivariable generating series; they are exactly the special values at non-positive integers of a complex multiple zeta-function. That equivalence governs both their analytic behavior and their later appearance in $L$0-adic interpolation.

A crucial distinction arises in the untwisted case. When $L$1, the original Euler–Zagier multiple zeta-function has infinitely many singular hyperplanes and is only meromorphic. The twisted theory avoids these singularities when all twists are nontrivial, while the untwisted case requires desingularization.

3. Relation to cyclotomic multiple Bernoulli numbers

A central development in the recent literature is the explicit relationship between TMBNs and cyclotomic multiple Bernoulli numbers (CMBNs). For “$L$2-steps,” the CMBNs are denoted

$L$3

and are characterized by the property that finite sums

$L$4

admit expansions in powers of $L$5 and $L$6 with coefficients given by these $L$7.

Theorem 2.8 of the 2026 paper gives an exact identity expressing

$L$8

as a finite sum over $L$9 with $p$0, integers $p$1 satisfying $p$2, and auxiliary indices $p$3. The terms involve the sign $p$4, powers $p$5, multinomial coefficients, Kronecker delta constraints in $p$6 and $p$7, and a final factor that is exactly a TMBN of lower depth (Fan, 5 Jul 2026).

Two special cases are singled out. The depth-one case is given explicitly in Corollary 2.9, and the constant-term case $p$8 is given in Corollary 2.10: $p$9

This CMBN-to-TMBN reduction is structurally important because Furusho–Jarossay expansions of $\xi$0-adic multiple $\xi$1-values initially involve CMBNs. The explicit formula shows that those coefficients can be rewritten in terms of TMBNs, and hence in terms of complex zeta-special values at non-positive integers. This suggests a precise bridge from cyclotomic harmonic data to complex special values through a finite combinatorial transform.

4. Desingularization, scaling, and parity phenomena

The twisted setting and the untwisted setting are linked by desingularization. When some $\xi$2, the generating-function definition still makes sense, but the corresponding untwisted multiple zeta-function is not entire. The desingularized function is defined by

$\xi$3

and becomes entire. Moreover, $\xi$4 is a finite $\xi$5-linear combination of shifted meromorphic zeta-functions, and its values at non-positive integers recover combinations of ordinary Bernoulli numbers (Furusho et al., 2013).

Several structural properties are emphasized in the literature. If all $\xi$6, then $\xi$7 is entire in the $\xi$8, so the special-value formula defining TMBNs by zeta-values is finite. When $\xi$9, or when all $$H(t;\xi)=\frac{1}{1-\xi e^t} =\sum_{n=-1}^{\infty} B_n(\xi)\frac{t^n}{n!}, \qquad (-1)!\equiv 1.$$0, one recovers the usual Bernoulli numbers up to minor shifts. The parameters $$H(t;\xi)=\frac{1}{1-\xi e^t} =\sum_{n=-1}^{\infty} B_n(\xi)\frac{t^n}{n!}, \qquad (-1)!\equiv 1.$$1 scale functorially: $$H(t;\xi)=\frac{1}{1-\xi e^t} =\sum_{n=-1}^{\infty} B_n(\xi)\frac{t^n}{n!}, \qquad (-1)!\equiv 1.$$2

A further symmetry is parity. From the generating-function identity

$$H(t;\xi)=\frac{1}{1-\xi e^t} =\sum_{n=-1}^{\infty} B_n(\xi)\frac{t^n}{n!}, \qquad (-1)!\equiv 1.$$3

one deduces

$$H(t;\xi)=\frac{1}{1-\xi e^t} =\sum_{n=-1}^{\infty} B_n(\xi)\frac{t^n}{n!}, \qquad (-1)!\equiv 1.$$4

In particular, all single twisted Bernoulli numbers of odd index vanish. This parity condition later reappears as a vanishing phenomenon for $$H(t;\xi)=\frac{1}{1-\xi e^t} =\sum_{n=-1}^{\infty} B_n(\xi)\frac{t^n}{n!}, \qquad (-1)!\equiv 1.$$5-adic multiple $$H(t;\xi)=\frac{1}{1-\xi e^t} =\sum_{n=-1}^{\infty} B_n(\xi)\frac{t^n}{n!}, \qquad (-1)!\equiv 1.$$6-functions and as a source of finite functional relations in lower depth.

These facts also clarify a frequent source of confusion: TMBNs should not be identified with untwisted multi-Bernoulli numbers without qualification. The twisted case is entire for nontrivial twists, whereas the untwisted case is accessed through a desingularized limit.

5. Explicit formulas in low depth

In depth one, the generating series immediately yields

$$H(t;\xi)=\frac{1}{1-\xi e^t} =\sum_{n=-1}^{\infty} B_n(\xi)\frac{t^n}{n!}, \qquad (-1)!\equiv 1.$$7

The notation $$H(t;\xi)=\frac{1}{1-\xi e^t} =\sum_{n=-1}^{\infty} B_n(\xi)\frac{t^n}{n!}, \qquad (-1)!\equiv 1.$$8 is used in the earlier paper, recovering Koblitz’s twisted Bernoulli numbers.

In depth two,

$$H(t;\xi)=\frac{1}{1-\xi e^t} =\sum_{n=-1}^{\infty} B_n(\xi)\frac{t^n}{n!}, \qquad (-1)!\equiv 1.$$9

Equivalently,

$B_{-1}(\xi)=-1,$0

The earlier explicit depth-two formula is

$B_{-1}(\xi)=-1,$1

This identity makes transparent how depth-two TMBNs decompose into combinations of depth-one twisted Bernoulli numbers with scale factors.

The untwisted desingularized depth-two case provides concrete comparison values: $B_{-1}(\xi)=-1,$2 producing rational values such as $B_{-1}(\xi)=-1,$3, $B_{-1}(\xi)=-1,$4, and $B_{-1}(\xi)=-1,$5 (Furusho et al., 2013). These examples show that the twisted theory is not isolated; it controls, after desingularization, the untwisted special-value formulas as well.

6. $B_{-1}(\xi)=-1,$6-adic multiple $B_{-1}(\xi)=-1,$7-functions and cyclotomic multiple harmonic values

For a prime $B_{-1}(\xi)=-1,$8, the $B_{-1}(\xi)=-1,$9-adic multiple $\xi=1$0-function is defined by

$\xi=1$1

where

$\xi=1$2

At non-positive integers, TMBNs are precisely the interpolation values of these functions. In the foundational formulation,

$\xi=1$3

so finite sums of TMBNs control the special values.

For positive integers, Furusho–Jarossay express the normalized values in terms of cyclotomic multiple harmonic values (CMHVs). If $\xi=1$4 and $\xi=1$5,

$\xi=1$6

Their theorem states that

$\xi=1$7

expands as a sum over data $\xi=1$8, cyclotomic parameters $\xi=1$9, and $B_n(1)=-\frac{B_{n+1}}{n+1}\qquad (n\ge 0).$0, with coefficients containing

$B_n(1)=-\frac{B_{n+1}}{n+1}\qquad (n\ge 0).$1

and CMHVs $B_n(1)=-\frac{B_{n+1}}{n+1}\qquad (n\ge 0).$2 (Fan, 5 Jul 2026).

The 2026 reformulation substitutes the explicit CMBN-to-TMBN identity together with

$B_n(1)=-\frac{B_{n+1}}{n+1}\qquad (n\ge 0).$3

and obtains a fully explicit expansion of the $B_n(1)=-\frac{B_{n+1}}{n+1}\qquad (n\ge 0).$4-adic multiple $B_n(1)=-\frac{B_{n+1}}{n+1}\qquad (n\ge 0).$5-values in terms of generalized Euler–Zagier–Lerch zeta-special values, combinatorial coefficients $B_n(1)=-\frac{B_{n+1}}{n+1}\qquad (n\ge 0).$6, and cyclotomic multiple harmonic values. The coefficients are therefore expressible directly by special values of complex functions at tuples of non-positive integers.

7. Congruences, vanishing, and polylogarithmic connections

The arithmetic theory includes a multiple analogue of Kummer congruences. Under the congruence condition

$B_n(1)=-\frac{B_{n+1}}{n+1}\qquad (n\ge 0).$7

the corresponding $B_n(1)=-\frac{B_{n+1}}{n+1}\qquad (n\ge 0).$8-adic multiple $B_n(1)=-\frac{B_{n+1}}{n+1}\qquad (n\ge 0).$9-values satisfy

$L$00

Since each such $L$01 is a finite sum of TMBNs, this is equivalently a congruence among finite linear combinations of TMBNs. An explicit depth-two version is written out in Example 4.7 (Furusho et al., 2013).

The parity symmetry has a $L$02-adic counterpart. The vanishing property of the Kubota–Leopoldt $L$03-adic $L$04-functions with odd characters extends to the multiple setting, yielding functional relations among $L$05-adic multiple $L$06-functions. In the complex-analytic language, these relations reflect parity-driven cancellation among higher-depth TMBNs.

At positive integers, the same framework connects to $L$07-adic twisted multiple polylogarithms via Coleman integration. Theorems 5.9 and 5.13 show that

$L$08

where $L$09 are rigid twisted multiple polylogarithms and $L$10 their Coleman-integral realizations. In this sense, TMBNs sit at the intersection of complex multiple-zeta special values, $L$11-adic interpolation, congruence theory, harmonic sums, and $L$12-adic polylogarithmic periods.

Taken together, these results present TMBNs as a coherent multivariable extension of classical Bernoulli numbers. Their generating-function definition, exact realization as special zeta-values, explicit relation to cyclotomic multiple Bernoulli numbers, parity and desingularization phenomena, and role in the special values of $L$13-adic multiple $L$14-functions form the present core of the theory.