--- title: Twisted Multiple Bernoulli Numbers (TMBNs) url: https://www.emergentmind.com/topics/twisted-multiple-bernoulli-numbers-tmbns type: topic --- # Twisted Multiple Bernoulli Numbers (TMBNs) 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 [2607.04086]. 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 [1309.3982]. ## 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 \(r\), with \[ \xi_1,\dots,\xi_r\in \mu_\infty\setminus\{1\},\qquad \gamma_1,\dots,\gamma_r\in \mathbb C,\ \Re(\gamma_j)>0, \] the generating function is \[ H_r\bigl((t_j);(\xi_j);(\gamma_j)\bigr) = \prod_{j=1}^r \frac{1}{1-\xi_j\exp\!\bigl(\gamma_j\sum_{k=j}^r t_k\bigr)} = \sum_{n_1,\dots,n_r\ge 0} B\bigl((n_j);(\xi_j);(\gamma_j)\bigr) \prod_{k=1}^r\frac{t_k^{n_k}}{n_k!}. \] The coefficient \[ B\bigl((n_j);(\xi_j);(\gamma_j)\bigr) \] is the depth-\(r\) twisted multiple Bernoulli number. An earlier notation writes the same family via \[ \mathfrak H_r\bigl((t_j);(\xi_j);(\gamma_j)\bigr) = \prod_{j=1}^r\frac{1}{1-\xi_j\exp\!\bigl(\gamma_j\sum_{k=j}^r t_k\bigr)} = \sum_{n_1,\dots,n_r=-1}^{\infty} (n_1,\dots,n_r;\xi_1,\dots,\xi_r;\gamma_1,\dots,\gamma_r) \frac{t_1^{n_1}\cdots t_r^{n_r}}{n_1!\cdots n_r!}, \] so that the coefficient \[ (n_1,\dots,n_r;\xi_j;\gamma_j) \] is the TMBN of index \((n_1,\dots,n_r)\), twists \(\xi_j\), and scales \(\gamma_j\) [1309.3982]. These definitions already exhibit two characteristic features. First, the twisting data are cyclotomic. Second, the variables \(t_j\) enter through nested sums \(\sum_{k=j}^r t_k\), so the depth-\(r\) 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 \[ \zeta_r\bigl((s_j);(\xi_j);(\gamma_j)\bigr) = \sum_{m_1,\dots,m_r\ge 1} \Bigl(\prod_{j=1}^r \xi_j^{m_j}\Bigr) \prod_{k=1}^r \bigl(m_1\gamma_1+\cdots+m_k\gamma_k\bigr)^{-s_k}, \] which converges in the region \[ \Re(s_{r-k+1}+\cdots+s_r)>k \qquad (1\le k\le r). \] If \(\xi_j\neq 1\) for all \(j\), this function admits analytic continuation to all of \(\mathbb C^r\), and indeed is entire. At tuples of non-positive integers one has the exact special-value formula \[ \zeta_r\bigl((-n_1,\dots,-n_r);(\xi_j);(\gamma_j)\bigr) = (-1)^{r+n_1+\cdots+n_r} \,(n_1,\dots,n_r;\xi_1^{-1},\dots,\xi_r^{-1};\gamma_1,\dots,\gamma_r), \] or, in the notation of the more recent paper, \[ \zeta_r\bigl((-n_j);(\xi_j);(\gamma_j)\bigr) = (-1)^{\,r+n_1+\cdots+n_r}\, B\bigl((n_j);(\xi_j^{-1});(\gamma_j)\bigr). \] 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 \(p\)-adic interpolation. A crucial distinction arises in the untwisted case. When \(\xi_j=1\), 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 “\(\kappa\)-steps,” the CMBNs are denoted \[ B_{l,\xi}^{\bigl((l_i)_r;(\epsilon_i)_r;(\kappa_i)_{r-1}\bigr)} \in \mathbb Z[\mu_c], \] and are characterized by the property that finite sums \[ S_{(\kappa_i),h}((l_i);(\epsilon_i)) \] admit expansions in powers of \(h\) and \(\xi\) with coefficients given by these \(B_{l,\xi}\). Theorem 2.8 of the 2026 paper gives an exact identity expressing \[ B_{l,\xi}^{\bigl((l_i);(\epsilon_i);(\kappa_i)\bigr)} \] as a finite sum over \(\mathbf e=(e_1,\dots,e_r)\in\{0,1\}^r\) with \(e_1=0\), integers \(m_{i,j}\ge 0\) satisfying \(\sum_{i\le j}m_{i,j}\le l_j\), and auxiliary indices \(i_k=0,\dots,1-e_k\). The terms involve the sign \((-1)^{\sum(e_k+i_k)}\), powers \((\kappa_{k-1}+i_k)^{\sum_{j\ge k}m_{k,j}}\), multinomial coefficients, Kronecker delta constraints in \(l\) and \(\xi\), and a final factor that is exactly a TMBN of lower depth [2607.04086]. Two special cases are singled out. The depth-one case is given explicitly in Corollary 2.9, and the constant-term case \(l=0\) is given in Corollary 2.10: \[ B_{0,\xi}^{\bigl((l_i);(\epsilon_i);(\kappa_i)\bigr)} = \sum_{\substack{(e_i)\in\{0,1\}^r,\;e_1=0}} \ \sum_{i_k=0}^{1-e_k} (-1)^{\sum(e_k+i_k)} \,B\!\bigl( l_1\circ_{e_2}\cdots\circ_{e_r}l_r\;;\; \epsilon_1^{-1}(1-e_1)\circ\cdots\circ\epsilon_r^{-1}(1-e_r) \bigr) \; \delta_{\xi,\prod_k\epsilon_k^{-(\kappa_{k-1}+i_k)}}. \] This CMBN-to-TMBN reduction is structurally important because Furusho–Jarossay expansions of \(p\)-adic multiple \(L\)-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 \(\xi_j=1\), the generating-function definition still makes sense, but the corresponding untwisted multiple zeta-function is not entire. The desingularized function is defined by \[ \zeta_r^{\rm des}\bigl((s_j);(\gamma_j)\bigr) = \lim_{c\to 1}\frac{(-1)^r}{(c-1)^r} \sum_{\substack{\xi_j^c=1\\ \xi_j\neq 1}} \zeta_r\bigl((s_j);(\xi_j);(\gamma_j)\bigr), \] and becomes entire. Moreover, \(\zeta_r^{\rm des}\) is a finite \(\mathbb Q\)-linear combination of shifted meromorphic zeta-functions, and its values at non-positive integers recover combinations of ordinary Bernoulli numbers [1309.3982]. Several structural properties are emphasized in the literature. If all \(\xi_j\neq 1\), then \(\zeta_r((s_j);(\xi_j);(\gamma_j))\) is entire in the \(s_j\), so the special-value formula defining TMBNs by zeta-values is finite. When \(r=1\), or when all \(\xi_j=1\), one recovers the usual Bernoulli numbers up to minor shifts. The parameters \(\gamma_j\) scale functorially: \[ B\bigl((n_j);(\xi_j);(\alpha\,\gamma_j)\bigr) = \alpha^{n_1+\cdots+n_r} \,B\bigl((n_j);(\xi_j);(\gamma_j)\bigr). \] A further symmetry is parity. From the generating-function identity \[ \widetilde{\mathcal H}_r(-t_j;\gamma_j;c)=(-1)^r\widetilde{\mathcal H}_r(t_j;\gamma_j;c), \] one deduces \[ (n_1,\dots,n_r;\gamma_j;c)=0 \qquad\text{whenever}\qquad n_1+\cdots+n_r\not\equiv r \pmod 2. \] In particular, all single twisted Bernoulli numbers of odd index vanish. This parity condition later reappears as a vanishing phenomenon for \(p\)-adic multiple \(L\)-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 \[ B_{-1}(\xi)=-1,\qquad B_0(\xi)=\frac{1}{1-\xi},\qquad B_1(\xi)=\frac{\xi}{(1-\xi)^2},\qquad B_2(\xi)=\frac{\xi(\xi+1)}{(1-\xi)^3},\ \dots \] The notation \({}_n(\xi)=(n;\xi;1)\) is used in the earlier paper, recovering Koblitz’s twisted Bernoulli numbers. In depth two, \[ H_2(t_1,t_2;\xi_1,\xi_2;\gamma_1,\gamma_2) = \frac{1}{ \bigl(1-\xi_1e^{\gamma_1(t_1+t_2)}\bigr) \bigl(1-\xi_2e^{\gamma_2 t_2}\bigr)} = \sum_{n_1,n_2\ge 0} B\bigl((n_1,n_2);(\xi_1,\xi_2);(\gamma_1,\gamma_2)\bigr) \frac{t_1^{n_1}}{n_1!}\frac{t_2^{n_2}}{n_2!}. \] Equivalently, \[ B\bigl((n_1,n_2);(\xi_1,\xi_2);(\gamma_1,\gamma_2)\bigr) = (-1)^{2+n_1+n_2}\, \zeta_2\!\bigl((-n_1,-n_2);(\xi_1^{-1},\xi_2^{-1});(\gamma_1,\gamma_2)\bigr). \] The earlier explicit depth-two formula is \[ (k,l;\xi_1,\xi_2;\gamma_1,\gamma_2) = \sum_{j=0}^{l}\binom{l}{j}\; {}_{k+j}(\xi_1)\,{}_{l-j}(\xi_2)\, \gamma_1^{k+j}\gamma_2^{l-j}. \] 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: \[ \zeta_2^{\rm des}(-k,-l;1,1) = (-1)^{k+l}\sum_{j=0}^{l}\binom{l}{j}\,B_{k+j+1}\,B_{l-j+1}, \] producing rational values such as \(1/18\), \(1/8\), and \(\zeta(3)-\zeta(4)\) [1309.3982]. These examples show that the twisted theory is not isolated; it controls, after desingularization, the untwisted special-value formulas as well. ## 6. \(p\)-adic multiple \(L\)-functions and cyclotomic multiple harmonic values For a prime \(p\nmid c\), the \(p\)-adic multiple \(L\)-function is defined by \[ L_{p,r}\bigl((s_i)_r;(\omega^{k_i})_r;(1)_r;c\bigr) = \int_{(p\mathbb Z_p)^r{}'} \prod_{j=1}^r \langle x_1+\cdots+x_j\rangle^{-s_j}\, \omega^{k_j-1}(x_1+\cdots+x_j)\; d\widetilde m_c(x_1)\cdots d\widetilde m_c(x_r), \] where \[ \widetilde m_c=\sum_{\substack{\xi^c=1\\ \xi\ne 1}} m_\xi. \] At non-positive integers, TMBNs are precisely the interpolation values of these functions. In the foundational formulation, \[ L_{p,r}(-n_j;\omega^{n_j};\gamma_j;c) = \sum_{\xi_j^c\ne 1} (n_j;\xi_j;\gamma_j) +\sum_{d\ge 1}(-1/p)^d\,(\text{lower depth terms}), \] 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 \(\epsilon_i\in\mu_c\) and \(n_i\in\mathbb N\), \[ H_m\bigl((n_i);(\epsilon_i)\bigr) = \sum_{0