{
  "Title": "Broadhurst–Mellit determinants of Bessel moments",
  "Definition": "For an integer $n\\geq3$, put $m=\\lfloor(n-1)/2\\rfloor$ and $B_n(a,r)=\\int_0^\\infty I_0(t)^aK_0(t)^{n-a}t^r\\,dt$, where $I_0$ and $K_0$ are the standard order-zero modified Bessel functions on the positive real axis. The Broadhurst–Mellit determinant is $D_n=\\det[B_n(a,2b-1)]_{a,b=1}^{m}$, with both rows and columns in increasing order CITE{Zhou}.",
  "Parameters": {
    "n": {
      "type": "Z",
      "title": "total number of Bessel factors",
      "display": "$n$",
      "constraints": "$n\\geq3$"
    }
  },
  "Comments": {
    "moments": "The matrix entries are HREF{Bessel_moments}[Bessel moments], with $B_n(a,r)=\\operatorname{IKM}(a,n-a;r)$. All Bessel factors have argument $t$ and common unit scale. No loop factors, powers of $\\pi$ or factorials are absorbed into the moments.",
    "convergence": "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 $n>2a$.",
    "notation": "In CITE{Zhou}, 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 CITE{Broadhurst} for the even formula with $m=n/2$.",
    "status": "Broadhurst states the product evaluations as Conjectures 4 and 7 in CITE{Broadhurst}. Zhou proves both for arbitrary matrix order in Section 4 of CITE{Zhou}; the full text is available in CITE{ZhouText}. The signed determinants in the stated row and column order are positive."
  },
  "References": {
    "Zhou": {
      "bib": "Yajun Zhou, Wrońskian factorizations and Broadhurst–Mellit determinant formulae (2018), equations (1.3)–(1.6) and Section 4.",
      "arxiv": "1711.01829"
    },
    "Broadhurst": {
      "bib": "David Broadhurst, Feynman integrals, L-series and Kloosterman moments (2016), Sections 7.5 and 7.9, equations (147)–(149) and (179)–(180).",
      "arxiv": "1604.03057"
    }
  },
  "Links": {
    "ZhouText": {
      "title": "Full text: Wrońskian factorizations and Broadhurst–Mellit determinant formulae",
      "url": "https://arxiv.org/html/1711.01829v2"
    }
  },
  "Tags": [
    "integral",
    "period",
    "special values",
    "physics",
    "number theory"
  ],
  "Data properties": {
    "type": "R",
    "rigour": "proven",
    "complete": "no",
    "complete-note": "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",
    "digits": "100",
    "rigour details": "The proved product formulas CITE{odd} and CITE{even} 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.\n\nAn 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 HREF{Bessel_moments}, within the precision of those agreement-checked moments.\n\nFor 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."
  },
  "Display properties": {
    "number-header": "$D_n$"
  },
  "Numbers": {
    "3": "0.6045997880780726168646927525473852440946887493642468585232949784627077270421179612280416627373533896",
    "4": "0.6168502750680849136771556874922594459571062129525494141508343360137528014012003276876108377324095145",
    "5": "0.2134876323745788358336868409316402991742383441758780730789046939834139853018021910794721787110552704",
    "6": "0.1691130052673653424243755775845574848085548362373010793254650336631438462792138822745171306158906121",
    "7": "0.05835380581641647181488152843914470246941132696373569296708252897835656170849100537189142052607239641",
    "8": "0.03016322776816868466418131352712874665799833765551593364750573290210377026186426892609743858025523862",
    "9": "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"
  },
  "Keywords": [
    "Broadhurst Mellit determinants",
    "Bessel moment determinants",
    "IKM",
    "equal-mass Bessel periods",
    "sunrise integrals"
  ],
  "Formulas": {
    "odd": "For $k\\geq1$, $D_{2k+1}=\\displaystyle\\prod_{j=1}^{k}\\frac{(2j)^{k-j}\\pi^j}{\\sqrt{(2j+1)^{2j+1}}}$ CITE{Zhou}, equation (1.4). The constant $\\pi$ is tabulated in HREF{Pi}.",
    "even": "For $k\\geq1$, $D_{2k+2}=\\displaystyle\\frac{2\\pi^{(k+1)^2/2}}{\\Gamma((k+1)/2)}\\prod_{j=1}^{k+1}\\frac{(2j-1)^{k+1-j}}{(2j)^j}$ CITE{Zhou}, equation (1.6). Here $\\Gamma$ is the HREF{Values_of_the_Gamma_function_at_rational_numbers}[Euler gamma function].",
    "initial": "$D_3=\\pi/(3\\sqrt3)$ and $D_4=\\pi^2/16$. These one-by-one determinants are the moments HREF{Bessel_moments#1,2,1}[$\\operatorname{IKM}(1,2;1)$] and HREF{Bessel_moments#1,3,1}[$\\operatorname{IKM}(1,3;1)$], respectively."
  },
  "Similar tables": [
    {
      "table": "HREF{Bessel_moments}[Bessel moments]",
      "relation": "holds individual matrix entries with the same unit-scale normalization; the one-by-one determinants are its moments with parameters $(1,2,1)$ and $(1,3,1)$"
    },
    {
      "table": "HREF{Pi}[$\\pi$]",
      "relation": "the circle constant occurring in both parity product formulas"
    },
    {
      "table": "HREF{Values_of_the_Gamma_function_at_rational_numbers}[Gamma values at rational arguments]",
      "relation": "the gamma values at $(k+1)/2$ occurring in the even-factor product formula"
    }
  ],
  "Programs": {
    "python": {
      "language": "Python",
      "code": "from mpmath import mp\nmp.dps = 120\ndef D(n):\n    if not isinstance(n, int) or n < 3:\n        raise ValueError(\"n must be an integer at least 3\")\n    k = (n - 1) // 2\n    if n % 2:\n        return mp.fprod(mp.mpf(2*j)**(k-j) * mp.pi**j /\n                        mp.sqrt(mp.mpf(2*j+1)**(2*j+1))\n                        for j in range(1, k+1))\n    return (2 * mp.pi**(mp.mpf((k+1)**2)/2) /\n            mp.gamma(mp.mpf(k+1)/2) *\n            mp.fprod(mp.mpf(2*j-1)**(k+1-j)/mp.mpf(2*j)**j\n                     for j in range(1, k+2)))\nfor n in range(3, 23):\n    print(n, mp.nstr(D(n), 100))"
    }
  }
}