--- title: Kaneko’s Formula in Number Theory and Beyond url: https://www.emergentmind.com/topics/kaneko-s-formula type: topic --- # Kaneko’s Formula in Number Theory and Beyond Searching arXiv for recent and foundational papers on “Kaneko’s formula” across number theory and related contexts. Kaneko’s formula is not a uniquely fixed expression across mathematics. In the cited literature, the term is used for several distinct identities associated with Masanobu Kaneko: most prominently, a singular-moduli formula for the Fourier coefficients of the elliptic modular function \(j(\tau)\); in a different arithmetic direction, a factorization phenomenon for higher derivatives of cyclotomic polynomials at \(x=1\); in enumerative combinatorics, a recursion for poly-Bernoulli numbers; and, in random matrix theory, a Selberg-type integral identity expressing certain \(\beta\)-ensemble integrals in terms of multivariate orthogonal polynomials [2507.21514] [2305.00765] [1510.05765] [2510.04422]. ## 1. Terminological scope The cited arXiv literature uses the expression “Kaneko’s formula” for several non-equivalent formulas. The most established usage is the level-\(1\) singular-moduli identity for \(j(\tau)\), but later papers also attach the label to formulas in cyclotomic, combinatorial, and Selberg-integral settings. | Context | Representative identity | arXiv paper | |---|---|---| | Singular moduli and \(j(\tau)\) | \(2n\,c(n)=\sum_{r\in\mathbb{Z}}\mathbf{t}_2(4n-r^2)\) | [2507.21514] | | Cyclotomic polynomials | \(F_{2k+1}(x_1,\dots,x_{2k+1})=(x_1-k)\,G_{2k+1}\) | [2305.00765] | | Poly-Bernoulli numbers | negative-index recursion derived combinatorially | [1510.05765] | | \(\beta\)-Laguerre/\(\beta\)-Jacobi ensembles | Selberg-type integral equals a multivariate orthogonal polynomial | [2510.04422] | A plausible implication is that “Kaneko’s formula” functions less as a universally standardized title than as a field-dependent shorthand for a structurally important identity first isolated by Kaneko or arising in work with Kaneko. ## 2. Singular moduli and the Fourier coefficients of \(j(\tau)\) In modular-form theory, Kaneko’s formula refers to an arithmetic expression for the Fourier coefficients of the elliptic modular function \[ j(\tau)=q^{-1}+744+\sum_{n=1}^{\infty}c(n)q^n,\qquad q=e^{2\pi i\tau}. \] For \(m\ge 1\), one defines polynomials \(j_m(\tau)\in \mathbb{C}[j]\) by the principal-part condition \[ j_m(\tau)=q^{-m}+O(q), \] and then the traces of singular moduli \[ \mathbf{t}_m(d)\coloneqq \sum_{Q\in\mathcal{Q}_d/\Gamma}\frac{1}{|\Gamma_Q|}\,j_m(\alpha_Q), \] where \(\mathcal{Q}_d\) is the set of positive definite integral binary quadratic forms of discriminant \(-d\), \(\alpha_Q\in\mathbb{H}\) is the CM point associated with \(Q\), and \(\Gamma_Q\) is the stabilizer in \(\Gamma=\operatorname{PSL}_2(\mathbb{Z})\) [2507.21514]. The form taken as the main statement in the recent survey is: \[ \sum_{r\in\mathbb{Z}} \mathbf{t}_2(4n-r^2)=2n\,\mathrm{Coeff}_{q^n}(j)=2n\,c(n), \qquad n\ge 1, \] with the auxiliary values \[ \mathbf{t}_2(0)=6,\qquad \mathbf{t}_2(-1)=-1,\qquad \mathbf{t}_2(-4)=-2, \] and \(\mathbf{t}_2(d)=0\) for all other integers \(d\) [2507.21514]. Equivalently, \[ c(n)=\frac{1}{2n}\sum_{r\in\mathbb{Z}}\mathbf{t}_2(4n-r^2). \] The original experimental formula discovered by Kaneko was written in terms of \(\mathbf{t}_1(d)\): \[ \mathrm{Coeff}_{q^n}(j) = \frac{1}{n}\sum_{r\in\mathbb{Z}}\left\{ \mathbf{t}_1(n-r^2) -\frac{(-1)^{n+r}}{4}\,\mathbf{t}_1(4n-r^2) +\frac{(-1)^r}{4}\,\mathbf{t}_1(16n-r^2) \right\}. \] The survey explains that this is equivalent to the \(\mathbf{t}_2\)-formula via the Hecke relation \[ \mathbf{t}_2(d)=\mathbf{t}_1(4d)+\left(\frac{-d}{2}\right)\mathbf{t}_1(d)+2\,\mathbf{t}_1(d/4) \] [2507.21514]. Conceptually, the formula is derived from Zagier’s modularity theorem for the trace-generating Jacobi form \[ g_m(\tau,z)\coloneqq \sum_{n\gg -\infty}\sum_{r\in\mathbb{Z}} \mathbf{t}_m(4n-r^2)\,q^n\zeta^r\in J_{2,1}^!. \] For \(m=2\), evaluating at \(z=0\) gives a weight-\(2\) weakly holomorphic modular form with principal part \(-2q^{-1}\), and comparison with \[ Dj(\tau)=q\frac{d}{dq}j(\tau)=-q^{-1}+\sum_{n\ge 1}n\,c(n)q^n \] shows that \(g_2(\tau,0)=2Dj(\tau)\), because the difference lies in \(M_2(\mathrm{SL}_2(\mathbb{Z}))\), which is zero [2507.21514]. The formula is concrete enough to be checked numerically at \(n=1\). The survey records \[ \mathbf{t}_2(3)=53256,\qquad \mathbf{t}_2(4)=287244, \] and then \[ \mathbf{t}_2(4)+2\mathbf{t}_2(3)+12=393768=2\cdot 196884, \] matching \(2\,\mathrm{Coeff}_{q}(j)\) [2507.21514]. ## 3. Higher-level generalizations and Eichler–Selberg relations Kaneko’s singular-moduli formula admits higher-level analogues for genus-zero McKay–Thompson series and for the Hecke system \[ j_m(\tau):=mT_m\bigl(j(\tau)-744\bigr). \] For square-free levels \(N\in\{2,3,5,6,7,10,13\}\), Matsusaka derives formulas for the Fourier coefficients of the Hauptmoduln \(j_N(\tau)\) and \(j_N^*(\tau)\) in terms of CM traces \(t_2^{(N)}(d)\) and \(t_2^{(N^*)}(d)\), together with explicit divisor-sum corrections [1703.10115]. At level \(1\), the same structural pattern can be written as \[ 2n c_n=\sum_{r\in\mathbb{Z}} t_1(4n-r^2), \] with \(c_n\) the coefficients of \(j(\tau)-744\), where \[ t_m(d)=\sum_{Q\in\mathcal{Q}_d/\mathrm{PSL}_2(\mathbb{Z})} \frac{1}{|\Gamma_Q|}\,\varphi_m\!\big(j(\alpha_Q)\big), \] and \(\varphi_m(j(\tau))=q^{-m}+O(q)\) [1703.10115]. The higher-level formulas replace \(j\) by \(j_N\) or \(j_N^*\), and the level-\(1\) trace functions by congruence-restricted CM traces. Structurally, they express \[ 2n(\text{Fourier coefficient}) = \sum_{r\in\mathbb{Z}}(\text{trace at }4n-r^2) + (\text{Eisenstein correction}) \] [1703.10115]. A more recent extension places Kaneko’s formulas inside an Eichler–Selberg framework for singular moduli. For \[ j_m(\tau)=q^{-m}+\sum_{n\ge 1}c_m(n)q^n=mT_m\bigl(j(\tau)-744\bigr), \] and \[ t_m(d):=\sum_{Q\in\mathcal{Q}_d/\Gamma}\frac{j_m(\alpha_Q)}{\#\Gamma_Q}, \] the generating functions \[ G_{m,\nu}(\tau)\coloneqq -\frac12\sum_{n\gg -\infty}\sum_{r\in\mathbb{Z}} p_{2\nu+2}(r,n)\,t_m(4n-r^2)\,q^n \] lie in \(M^!_{2\nu+2}\) for \(\nu\ge 0\), \(m\ge 1\) [2406.14280]. For \(\nu=0\), \[ \sum_{r\in\mathbb{Z}} t_1(4n-r^2)=0,\qquad \sum_{r\in\mathbb{Z}} t_2(4n-r^2)=2n\,c_1(n), \] and these are identified as the \(m=1,2\) cases underlying Kaneko’s singular-moduli formulas [2406.14280]. For \(\nu\ge 1\), the same program produces new Eichler–Selberg trace formulas in which the traces of \(j_m(\tau)\) singular moduli replace Hurwitz–Kronecker class numbers. The resulting identities involve a new term assembled from values of symmetrized shifted convolution \(L\)-functions \[ \widehat{L}(f,m;s) = \sum_{n\ge 1}\frac{c_f(n)c_f(n+m)}{n^s} - \sum_{n\ge 1}\frac{c_f(n)c_f(n-m)}{n^s}, \] where \(f\) ranges over normalized Hecke eigenforms in \(S_{2\nu+2}\) [2406.14280]. This places Kaneko’s original formula at the initial, weight-\(2\) end of a broader hierarchy. ## 4. Cyclotomic polynomials and the Akiyama–Kaneko factorization In the cyclotomic setting, “Kaneko’s formula” refers not to the modular \(j\)-function but to explicit formulas and factorization phenomena for the higher derivatives of cyclotomic polynomials at \(x=1\). The starting point is Lehmer’s statement that for each \(k\ge 1\) there exists a polynomial \[ F_k(x_1,\dots,x_k)\in \mathbb{Q}[x_1,\dots,x_k] \] such that \[ \frac{\Phi_n^{(k)}(1)}{\Phi_n(1)} = F_k\!\left(\frac{\varphi(n)}{2},\frac{J_2(n)}{4},\dots,\frac{J_k(n)}{2k}\right), \] where \[ J_m(n)=n^m\prod_{p\mid n}\left(1-p^{-m}\right) =\sum_{d\mid n}\mu\!\left(\frac{n}{d}\right)d^m \] is Jordan’s totient function [2305.00765]. Akiyama and Kaneko observed empirically that the odd-index polynomials \(F_{2k+1}\) contain a simple linear factor. The conjecture proved in the paper is: \[ F_{2k+1}(x_1,\dots,x_{2k+1}) \text{ is divisible by }x_1-k \text{ in }\mathbb{Q}[x_1,\dots,x_{2k+1}]. \] The structural identity establishing this is \[ F_k(x_1,\dots,x_k) = (x_1)_k + 2\sum_{m=1}^{\lfloor k/2\rfloor} B_{2m}\binom{k}{2m}(x_1-m)_{k-2m}\,\Omega_m(x_2,\dots,x_{2m}), \] where \((x)_n=x(x-1)\cdots(x-n+1)\), \(B_{2m}\) are Bernoulli numbers, and \(\Omega_m\) is defined by a generating function [2305.00765]. For odd \(k=2r+1\), every term contains the factor \(x_1-r\), yielding the corollary \[ F_{2k+1}(x_1,\dots,x_{2k+1})=(x_1-k)\,G_{2k+1}(x_1,\dots,x_{2k+1}) \] for some \(G_{2k+1}\in\mathbb{Q}[x_1,\dots,x_{2k+1}]\) [2305.00765]. After the Lehmer substitution \(x_1=\varphi(n)/2\), the factor \(x_1-k\) becomes \((\varphi(n)-2k)/2\). This explains the “curious congruences” observed by Akiyama and Kaneko: 1. \(2\Phi_n^{(3)}(1)\) is divisible by \(\phi(n)-2\). 2. If \(k\ge 2\), then \(\Phi_n^{(2k+1)}(1)\) is divisible by \(\phi(n)-2k\). The paper further proves that for integers \(n\ge 3\) and \(1\le k<\phi(n)\), \[ F_{k,n}(x):= F_k\!\left(x,\frac{J_2(n)}{4},\dots,\frac{J_k(n)}{2k}\right)\in \mathbb{Z}[x], \] and derives refined congruences for the ratios \(\Phi_n^{(2k+1)}(1)/\Phi_n(1)\) [2305.00765]. In this usage, Kaneko’s formula is best understood as the odd-order factorization property of the Lehmer polynomials \(F_k\), together with their explicit arithmetic specialization. ## 5. Poly-Bernoulli numbers and Kaneko’s recursive formula In combinatorics, Kaneko introduced the poly-Bernoulli numbers \(B_n^{(k)}\) by the exponential generating function \[ \sum_{n=0}^{\infty} B_n^{(k)} \frac{x^n}{n!} = \frac{\operatorname{Li}_k(1-e^{-x})}{1-e^{-x}}, \qquad \operatorname{Li}_k(x)=\sum_{i=1}^{\infty}\frac{x^i}{i^k}. \] The paper emphasizes that \(B_n^{(1)}\) recovers the Bernoulli numbers, with the convention \(B_1^{(1)}=+\tfrac12\) [1510.05765]. For negative indices, Arakawa and Kaneko proved the Stirling-number expansion \[ B_n^{(-k)} = \sum_{m=0}^{\min\{n,k\}} m!\, \left\{\begin{matrix} n+1 \\ m+1 \end{matrix}\right\} \left\{\begin{matrix} k+1 \\ m+1 \end{matrix}\right\}, \] which immediately shows that \(B_n^{(-k)}\) is a nonnegative integer [1510.05765]. The same paper surveys several combinatorial models realizing this formula, including lonesum \(0\text{-}1\) matrices, Callan permutations, max-ascending permutations, Vesztergombi permutations, and acyclic orientations of \(K_{n,k}\) [1510.05765]. Its new interpretation uses \(T\)-free matrices. Let \(\mathcal{G}_{n}^{(k)}\) be the set of \(n\times k\) \(0\text{-}1\) matrices avoiding the configuration in which three \(1\)s form the upper-left, upper-right, and lower-left entries of a \(2\times 2\) submatrix. The paper proves \[ |\mathcal{G}_{n}^{(k)}|=B_n^{(-k)}, \] and then derives a direct combinatorial recursion: \[ B_n^{(-k)} = B_n^{(-(k-1))} + \sum_{j=1}^{n} \binom{n}{j-1} B_{n-(j-1)}^{(-(k-1))}. \] This is presented as a transparent combinatorial explanation of Kaneko’s recursive formula [1510.05765]. The same article records the symmetry \[ B_n^{(-k)}=B_k^{(-n)}, \] noting that it is immediate from the Stirling-number formula and from several of the combinatorial realizations [1510.05765]. In this context, Kaneko’s formula is a recursion in a two-parameter refinement of Bernoulli-number theory rather than a statement about modular forms or CM values. ## 6. Kaneko’s integral formula in \(\beta\)-ensemble theory A different usage appears in random matrix theory and multivariate orthogonal polynomials. For the Jacobi ensemble, Kaneko’s integral formula is stated as the identity \[ \int_{[0,1]^\nu}\prod_{j=1}^\nu\prod_{i=1}^n (x_i-y_j)\, w_{n,\gamma_1,\gamma_2}^{J,\beta}(x_1,\dots,x_n)\,dx_1\cdots dx_n = \frac{1}{\mathcal{Z}_{n,\gamma_1+\nu,\gamma_2}^{J,\beta}\, P_{n^{(\nu)},\tilde{\gamma}_1,\tilde{\gamma}_2}^{\tilde{\beta}}(y_1,\dots,y_\nu)}, \] with \[ w_{n,\gamma_1,\gamma_2}^{J,\beta}(x_1,\dots,x_n) = \prod_{i