Broadhurst–Mellit determinants of Bessel moments

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Numbers

$n$

$D_n$

0.6045997880780726168646927525473852440946887493642468585232949784627077270421179612280416627373533896

0.6168502750680849136771556874922594459571062129525494141508343360137528014012003276876108377324095145

0.2134876323745788358336868409316402991742383441758780730789046939834139853018021910794721787110552704

0.1691130052673653424243755775845574848085548362373010793254650336631438462792138822745171306158906121

0.05835380581641647181488152843914470246941132696373569296708252897835656170849100537189142052607239641

0.03016322776816868466418131352712874665799833765551593364750573290210377026186426892609743858025523862

0.01386176785965996163193380976675781153128090657863801241337148535438312157268548029085147922659232376

10:

0.004113441639368450993574812693601817608013442918510656448008421890872840229285340689414360993901799664

11:

0.003049575858839089305308289173121676237296736007395278116790782223571626830232654319398154624595043988

12:

0.0004693320106241962602194089947206197097283448222767702054796409589314114711232279576803426830530200135

13:

0.0006469014305458160054403589309851847217963859114150049737426826822267368117487521456990573085029230141

14:

0.00004746094484238513868243526491055253496771299191696105635994146268310237996541459271113533386050198397

15:

0.0001360552322099552981849783708428050971380009472639290790539881065226128955775849297418952205923045396

16:

0.000004427854768283115155952956427822604414619946140107378053384047947035354063876893578873239136950308262

17:

0.00002895602239325643711651519468379401437109825316729697312476839722799146877325928616954622876675911860

18:

3.925372199779295129819390921302406327519261381506031576521678473774242387075416755196923946112099863e-7

19:

0.000006334160122104528679405009705560641679605053022665081148311052375148770505700190174085526964709283785

20:

3.382561596139302444585772085666107864982289416427636816558374563424843245955433265868646062914043296e-8

21:

0.000001441843632561958685512639202758372937517380814594438612247965186913138453125455540975067716065371334

22:

2.884602599745443883885525961247021955667891660315882415893300051726149685829542072944187266499471640e-9

Definition

For $n\geq3$ and $m=\lfloor(n-1)/2\rfloor$, the Broadhurst–Mellit determinant is $D_n=\det[B_n(a,2b-1)]_{a,b=1}^{m}$, with rows and columns in increasing order [1].

Parameters

$n$

— total number of Bessel factors ($n\geq3$)

Formulas

(1)

For $k\geq1$, $D_{2k+1}=\displaystyle\prod_{j=1}^{k}\frac{(2j)^{k-j}\pi^j}{\sqrt{(2j+1)^{2j+1}}}$ [1], equation (1.4). The constant $\pi$ is tabulated in Pi.

(2)

For $k\geq1$, $D_{2k+2}=\displaystyle\frac{2\pi^{(k+1)^2/2}}{\Gamma((k+1)/2)}\prod_{j=1}^{k+1}\frac{(2j-1)^{k+1-j}}{(2j)^j}$ [1], equation (1.6). Here $\Gamma$ is the Euler gamma function.

(3)

$D_3=\pi/(3\sqrt3)$ and $D_4=\pi^2/16$. These one-by-one determinants are the moments $\operatorname{IKM}(1,2;1)$ and $\operatorname{IKM}(1,3;1)$, respectively.

(4)

For $n=5,6$, the vacuum determinant satisfies $V_n=n\mu_nD_n$, where $\mu_n=m(1+x_1+\cdots+x_{n-1})$ is the logarithmic Mahler measure. This follows by comparing equations (1.4), (1.6), (1.9) and (1.10) of [1]; it does not assert an analogous formula for the different eight-factor construction.

Comments

(5)

The unit-scale Bessel moment is $B_n(a,r)=\int_0^\infty I_0(t)^aK_0(t)^{n-a}t^r\,dt$, where $I_0$ and $K_0$ are the standard order-zero modified Bessel functions on the positive real axis. The moment is also denoted $\operatorname{IKM}(a,n-a;r)$. Every factor has argument $t$; no loop factors, powers of $\pi$ or factorials are absorbed into the moments.

(6)

Every integral converges ordinarily. Its integrand is $O(t^r(1+|\log t|)^{n-a})$ as $t\to0^+$ and $O(t^{r-n/2}e^{-(n-2a)t})$ as $t\to\infty$; here $r=2b-1\geq1$ and $n>2a$.

(7)

In [1], the odd case $D_{2k+1}$ is $\det\mathbf M_k$ and the even case $D_{2k+2}$ is $\det\mathbf N_k$. The symbol $D_n$ here indexes total factor count; Broadhurst uses $D_m$ in equation (180) of [2] for the even formula with $m=n/2$.

(8)

Broadhurst states the product evaluations as Conjectures 4 and 7 in [2]. Zhou proves both for arbitrary matrix order in Section 4 of [1]; the full text is available in [3]. The signed determinants in the stated row and column order are positive.

(9)

The vacuum determinants are different combinations of Bessel moments, not the individual entries of these matrices. The present determinant is defined for every integer $n\geq3$, and the proved product formulas apply throughout that family. The vacuum table selects three published matrices at $n=5,6,8$, so the two tables need not cover the same factor counts. The product-formula generator here does not use vacuum-determinant values.

Programs

(P1)

Python

from mpmath import mp
mp.dps = 120
def D(n):
    if not isinstance(n, int) or n < 3:
        raise ValueError("n must be an integer at least 3")
    k = (n - 1) // 2
    if n % 2:
        return mp.fprod(mp.mpf(2*j)**(k-j) * mp.pi**j /
                        mp.sqrt(mp.mpf(2*j+1)**(2*j+1))
                        for j in range(1, k+1))
    return (2 * mp.pi**(mp.mpf((k+1)**2)/2) /
            mp.gamma(mp.mpf(k+1)/2) *
            mp.fprod(mp.mpf(2*j-1)**(k+1-j)/mp.mpf(2*j)**j
                     for j in range(1, k+2)))
for n in range(3, 23):
    print(n, mp.nstr(D(n), 100))

References

[1]

Yajun Zhou, Wrońskian factorizations and Broadhurst–Mellit determinant formulae (2018), equations (1.3)–(1.6) and Section 4. (arXiv)

[2]

David Broadhurst, Feynman integrals, L-series and Kloosterman moments (2016), Sections 7.5 and 7.9, equations (147)–(149) and (179)–(180). (arXiv)

Links

[3]

Full text: Wrońskian factorizations and Broadhurst–Mellit determinant formulae

Similar tables

Bessel moments — holds individual matrix entries with the same unit-scale normalization; the one-by-one determinants are its moments with parameters $(1,2,1)$ and $(1,3,1)$

$\pi$ — the circle constant occurring in both parity product formulas

Gamma values at rational arguments — the gamma values at $(k+1)/2$ occurring in the even-factor product formula

Broadhurst–Mellit vacuum determinants — uses the same unit-scale moments in vacuum-containing matrices. For $n=5,6$, $V_n$ replaces the $a=1$ row of $D_n$ by $a=0$, keeping the $a=2$ row and columns $r=1,3$. At $n=8$, $V_8$ uses a two-by-two matrix with second column $B_8(a,3)-2B_8(a,5)$, whereas $D_8$ is three-by-three

Data properties

Entries are of type: real number

How well the digits are known: proven

Table is complete: no (it holds every total factor count $3\leq n\leq22$, covering matrix orders $1$ through $10$ for both parities. This initial prefix includes the scalar base cases, the two-by-two period relations and higher orders in one discrete family; order ten is a practical reference cutoff, not a mathematical boundary)

How they were obtained:

The proved product formulas (1) and (2) are evaluated with exact integer and rational prefactors and Sage/Arb real ball arithmetic. Each ball encloses the exact formula value and supports the 100 significant digits written here. The default computation uses 397 bits, including 64 extra working bits; the smallest measured relative accuracy across the 20 results exceeds 117 decimal digits. Every written interval contains a recomputation at 662 bits.

An independent mpmath computation at 180 working decimal digits uses Zhou's coupled determinant recursions, equations (4.1) and (4.2), starting from the one-by-one cases. Its largest relative discrepancy from the higher-precision product evaluation is below $7\times10^{-179}$. The two one-by-one and two two-by-two determinants also agree with the corresponding stored moments in Bessel moments, within the precision of those agreement-checked moments.

For every $3\leq n\leq22$, the defining moment matrix was independently integrated and its determinant evaluated at 70 and 100 working decimal digits. The largest relative discrepancies from the products were below $4\times10^{-67}$ and $2.4\times10^{-95}$, respectively. The nonzero controls $\int_0^\infty tK_0(t)\,dt=1$ and $\int_0^\infty tK_0(t)^2\,dt=1/2$ pass before the determinant checks. Bessel factors are evaluated in complex ball arithmetic at adaptively increased precision before their midpoints enter tanh-sinh quadrature. These integral checks are heuristic comparisons, not certificates of integration error; the rigorous stored-digit bounds come from the product formulas. The attached checks and numerical evidence give the quadrature cutoffs, precision records and determinant conditioning diagnostics.