Kaneko’s Formula in Number Theory and Beyond

• Kaneko’s Formula is a term for several distinct mathematical identities, including those in modular forms, cyclotomic polynomials, combinatorial recursions, and random matrix theory.
• In modular form theory, it expresses the Fourier coefficients of the elliptic modular function j(τ) through sums over singular moduli, providing deep insights into arithmetic properties.
• Generalizations extend the formula to higher-level modular functions, factorization properties for cyclotomic polynomials, recurrences for poly-Bernoulli numbers, and Selberg-type integral identities in β-ensembles.

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 (Matsusaka, 29 Jul 2025, 2305.00765, Bényi et al., 2015, Jeong, 6 Oct 2025).

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 $j(\tau)$	$2n\,c(n)=\sum_{r\in\mathbb{Z}}\mathbf{t}_2(4n-r^2)$	(Matsusaka, 29 Jul 2025)
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	(Bényi et al., 2015)
$\beta$-Laguerre/$\beta$-Jacobi ensembles	Selberg-type integral equals a multivariate orthogonal polynomial	(Jeong, 6 Oct 2025)

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 $x=1$0

In modular-form theory, Kaneko’s formula refers to an arithmetic expression for the Fourier coefficients of the elliptic modular function

$x=1$1

For $x=1$2, one defines polynomials $x=1$3 by the principal-part condition

$x=1$4

and then the traces of singular moduli

$x=1$5

where $x=1$6 is the set of positive definite integral binary quadratic forms of discriminant $x=1$7, $x=1$8 is the CM point associated with $x=1$9, and $\beta$0 is the stabilizer in $\beta$1 (Matsusaka, 29 Jul 2025).

The form taken as the main statement in the recent survey is: $\beta$2 with the auxiliary values

$\beta$3

and $\beta$4 for all other integers $\beta$5 (Matsusaka, 29 Jul 2025). Equivalently,

$\beta$6

The original experimental formula discovered by Kaneko was written in terms of $\beta$7: $\beta$8 The survey explains that this is equivalent to the $\beta$9-formula via the Hecke relation

$1$0

(Matsusaka, 29 Jul 2025).

Conceptually, the formula is derived from Zagier’s modularity theorem for the trace-generating Jacobi form

$1$1

For $1$2, evaluating at $1$3 gives a weight-$1$4 weakly holomorphic modular form with principal part $1$5, and comparison with

$1$6

shows that $1$7, because the difference lies in $1$8, which is zero (Matsusaka, 29 Jul 2025).

The formula is concrete enough to be checked numerically at $1$9. The survey records

$j(\tau)$0

and then

$j(\tau)$1

matching $j(\tau)$2 (Matsusaka, 29 Jul 2025).

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(\tau)$3

For square-free levels $j(\tau)$4, Matsusaka derives formulas for the Fourier coefficients of the Hauptmoduln $j(\tau)$5 and $j(\tau)$6 in terms of CM traces $j(\tau)$7 and $j(\tau)$8, together with explicit divisor-sum corrections (Matsusaka, 2017).

At level $j(\tau)$9, the same structural pattern can be written as

$j(\tau)$0

with $j(\tau)$1 the coefficients of $j(\tau)$2, where

$j(\tau)$3

and $j(\tau)$4 (Matsusaka, 2017). The higher-level formulas replace $j(\tau)$5 by $j(\tau)$6 or $j(\tau)$7, and the level-$j(\tau)$8 trace functions by congruence-restricted CM traces. Structurally, they express

$j(\tau)$9

(Matsusaka, 2017).

A more recent extension places Kaneko’s formulas inside an Eichler–Selberg framework for singular moduli. For

$2n\,c(n)=\sum_{r\in\mathbb{Z}}\mathbf{t}_2(4n-r^2)$0

and

$2n\,c(n)=\sum_{r\in\mathbb{Z}}\mathbf{t}_2(4n-r^2)$1

the generating functions

$2n\,c(n)=\sum_{r\in\mathbb{Z}}\mathbf{t}_2(4n-r^2)$2

lie in $2n\,c(n)=\sum_{r\in\mathbb{Z}}\mathbf{t}_2(4n-r^2)$3 for $2n\,c(n)=\sum_{r\in\mathbb{Z}}\mathbf{t}_2(4n-r^2)$4, $2n\,c(n)=\sum_{r\in\mathbb{Z}}\mathbf{t}_2(4n-r^2)$5 (Deng et al., 2024). For $2n\,c(n)=\sum_{r\in\mathbb{Z}}\mathbf{t}_2(4n-r^2)$6,

$2n\,c(n)=\sum_{r\in\mathbb{Z}}\mathbf{t}_2(4n-r^2)$7

and these are identified as the $2n\,c(n)=\sum_{r\in\mathbb{Z}}\mathbf{t}_2(4n-r^2)$8 cases underlying Kaneko’s singular-moduli formulas (Deng et al., 2024).

For $2n\,c(n)=\sum_{r\in\mathbb{Z}}\mathbf{t}_2(4n-r^2)$9, the same program produces new Eichler–Selberg trace formulas in which the traces of $F_{2k+1}(x_1,\dots,x_{2k+1})=(x_1-k)\,G_{2k+1}$0 singular moduli replace Hurwitz–Kronecker class numbers. The resulting identities involve a new term assembled from values of symmetrized shifted convolution $F_{2k+1}(x_1,\dots,x_{2k+1})=(x_1-k)\,G_{2k+1}$1-functions

$F_{2k+1}(x_1,\dots,x_{2k+1})=(x_1-k)\,G_{2k+1}$2

where $F_{2k+1}(x_1,\dots,x_{2k+1})=(x_1-k)\,G_{2k+1}$3 ranges over normalized Hecke eigenforms in $F_{2k+1}(x_1,\dots,x_{2k+1})=(x_1-k)\,G_{2k+1}$4 (Deng et al., 2024). This places Kaneko’s original formula at the initial, weight-$F_{2k+1}(x_1,\dots,x_{2k+1})=(x_1-k)\,G_{2k+1}$5 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 $F_{2k+1}(x_1,\dots,x_{2k+1})=(x_1-k)\,G_{2k+1}$6-function but to explicit formulas and factorization phenomena for the higher derivatives of cyclotomic polynomials at $F_{2k+1}(x_1,\dots,x_{2k+1})=(x_1-k)\,G_{2k+1}$7. The starting point is Lehmer’s statement that for each $F_{2k+1}(x_1,\dots,x_{2k+1})=(x_1-k)\,G_{2k+1}$8 there exists a polynomial

$F_{2k+1}(x_1,\dots,x_{2k+1})=(x_1-k)\,G_{2k+1}$9

such that

$\beta$0

where

$\beta$1

is Jordan’s totient function (2305.00765).

Akiyama and Kaneko observed empirically that the odd-index polynomials $\beta$2 contain a simple linear factor. The conjecture proved in the paper is: $\beta$3 The structural identity establishing this is

$\beta$4

where $\beta$5, $\beta$6 are Bernoulli numbers, and $\beta$7 is defined by a generating function (2305.00765). For odd $\beta$8, every term contains the factor $\beta$9, yielding the corollary

$\beta$0

for some $\beta$1 (2305.00765).

After the Lehmer substitution $\beta$2, the factor $\beta$3 becomes $\beta$4. This explains the “curious congruences” observed by Akiyama and Kaneko:

1. $\beta$5 is divisible by $\beta$6.
2. If $\beta$7, then $\beta$8 is divisible by $\beta$9.

The paper further proves that for integers $x=1$00 and $x=1$01,

$x=1$02

and derives refined congruences for the ratios $x=1$03 (2305.00765). In this usage, Kaneko’s formula is best understood as the odd-order factorization property of the Lehmer polynomials $x=1$04, together with their explicit arithmetic specialization.

5. Poly-Bernoulli numbers and Kaneko’s recursive formula

In combinatorics, Kaneko introduced the poly-Bernoulli numbers $x=1$05 by the exponential generating function

$x=1$06

The paper emphasizes that $x=1$07 recovers the Bernoulli numbers, with the convention $x=1$08 (Bényi et al., 2015).

For negative indices, Arakawa and Kaneko proved the Stirling-number expansion

$x=1$09

which immediately shows that $x=1$10 is a nonnegative integer (Bényi et al., 2015). The same paper surveys several combinatorial models realizing this formula, including lonesum $x=1$11 matrices, Callan permutations, max-ascending permutations, Vesztergombi permutations, and acyclic orientations of $x=1$12 (Bényi et al., 2015).

Its new interpretation uses $x=1$13-free matrices. Let $x=1$14 be the set of $x=1$15 $x=1$16 matrices avoiding the configuration in which three $x=1$17s form the upper-left, upper-right, and lower-left entries of a $x=1$18 submatrix. The paper proves

$x=1$19

and then derives a direct combinatorial recursion: $x=1$20 This is presented as a transparent combinatorial explanation of Kaneko’s recursive formula (Bényi et al., 2015).

The same article records the symmetry

$x=1$21

noting that it is immediate from the Stirling-number formula and from several of the combinatorial realizations (Bényi et al., 2015). 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 $x=1$22-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

$x=1$23

with

$x=1$24

$x=1$25

and $x=1$26 the multivariate Jacobi polynomial indexed by the square partition $x=1$27 (Jeong, 6 Oct 2025). The paper then obtains a Laguerre analogue by taking a Jacobi-to-Laguerre limit: $x=1$28

The significance of this identity in the paper is operational. When the exponent of $x=1$29 in the Laguerre or Jacobi weight is an integer, the smallest-eigenvalue CDF and density for the $x=1$30-Laguerre and $x=1$31-Jacobi ensembles can be rewritten as integrals with inserted factors $x=1$32, and Kaneko’s formula converts those integrals into explicit evaluations of multivariate Laguerre or Jacobi polynomials at scalar matrix arguments (Jeong, 6 Oct 2025).

For the Laguerre case, this yields

$x=1$33

and an analogous closed form for the density $x=1$34. The Jacobi case gives parallel formulas with multivariate Jacobi polynomials $x=1$35 (Jeong, 6 Oct 2025).

The paper then derives new differentiation formulas for these multivariate polynomials and, at $x=1$36, explicit rational solutions of the Painlevé V and VI equations governing the smallest eigenvalue distributions in the LUE and JUE. In this setting, Kaneko’s formula is an integral transform from generalized Selberg integrals to multivariate orthogonal polynomials, and its role is computational as well as structural (Jeong, 6 Oct 2025).