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{
"Title": "Bessel moments",
"Definition": "The Bessel moments CITE{Zhou} are $\\operatorname{IKM}(a,b;k)=\\int_0^\\infty t^k I_0(t)^aK_0(t)^b\\,dt$, where $I_0$ and $K_0$ are the standard order-zero modified Bessel functions on the positive real axis.",
"Parameters": {
"a": {
"type": "Z",
"title": "number of first-kind factors",
"constraints": "$a\\geq0$",
"display": "$a$"
},
"b": {
"type": "Z",
"title": "number of second-kind factors",
"constraints": "$b>a$",
"display": "$b$"
},
"k": {
"type": "Z",
"title": "moment power",
"constraints": "$k\\geq0$",
"display": "$k$"
}
},
"Tags": [
"integral",
"special values",
"special functions",
"physics",
"period"
],
"Links": {
"BBBGtext": {
"title": "Full text: Elliptic integral evaluations of Bessel moments",
"url": "https://arxiv.org/html/0801.0891v2"
}
},
"Data properties": {
"type": "R",
"rigour": "heuristic (agreement-checked)",
"digits": "100",
"complete": "no",
"complete-note": "it holds all 40 moments with $3\\leq a+b\\leq6$, $0\\leq a<b$ and $k=1,3,5,7$. These are the low-factor equal-mass families studied in CITE{BBBG}; the base measure $t\\,dt$ and increments by two come from the vacuum and sunrise integrals and mass differentiation. The factor and moment bounds are short prefixes of these discrete families",
"rigour details": "Direct tanh-sinh quadrature at 150 and 200 working decimal digits, with Arb evaluations of the Bessel factors, gives 100 significant stored digits. The worst relative discrepancy between the two runs was 6.064e-152. Arb midpoints enter the quadrature, so this is agreement evidence, not a certified integration enclosure. The upper endpoint is truncated at T=3*(working digits+60); the generator documents an explicit exponentially small tail bound. The integrand tends to zero at the lower endpoint for every selected moment.\n\nThe nonzero controls IKM(0,1;1)=1 and IKM(0,2;1)=1/2 were tested before the table values. All 14 applicable recurrence relations and 11 closed-form comparisons, including exact multinomial counts, were tested on the computed values. A separate quadrature uses mpmath Bessel functions, Gauss-Legendre integration and t=exp(x) at 60 working digits; it compares all 40 entries with a relative tolerance of 1e-50. These numerical checks are reproduced in checks.py.\n\nAfter submission, all 40 values were read back and found identical to the submitted decimal strings. The 25 identity checks and the independent quadrature were repeated on these stored values, and generator.verify(sample=None) matched all 40 entries."
},
"Display properties": {
"number-header": "$\\operatorname{IKM}(a,b;k)$",
"group parameters": [
[
"a",
"b"
],
[
"k"
]
]
},
"Numbers": {
"0,3": {
"1": "0.5859768096723647226503905722180692672738507524089640651665750472251652384887136635696521781241573958",
"3": "0.1146357462298196302005207629574256896984676698786187535554333963002203179849515514262029041655431944",
"5": "0.1300955220428411573379623373993394639863792878099444760482968560019583820884582348995813703603839498",
"7": "0.3819245022180162575370215230591742624177618932421785615533175844041234313493197810651735404930775110"
},
"1,2": {
"1": "0.6045997880780726168646927525473852440946887493642468585232949784627077270421179612280416627373533896",
"3": "0.8061330507707634891529236700631803254595849991523291446977266379502769693894906149707222169831378528",
"5": "7.165627117962342125803765956117158448529644436909592397313125670669128616795472133073086373183447581",
"7": "177.7075525254660847199333957117055295235351820353578914533655166325943896965277089002125420549495000"
},
"0,4": {
"1": "1.051799790264644999724770891322518741919363005797936521568237610924108430063023953913148889408168351",
"3": "0.07544994756616124993119272283062968547984075144948413039205940273102710751575598847828722235204208772",
"5": "0.03828740824078218737958726495360194958972131503659722818610395477929743815257297983700263911607365351",
"7": "0.05404952809545124938073450547566716931856676304535717352853462457924396764180389630458500116837878948"
},
"1,3": {
"1": "0.6168502750680849136771556874922594459571062129525494141508343360137528014012003276876108377324095145",
"3": "0.1542125687670212284192889218730648614892765532381373535377085840034382003503000819219027094331023786",
"5": "0.2698719953422871497337556132778635076062339681667403686909900220060168506130251433633297415079291626",
"7": "1.387913118903191055773600296857583753403488979143236181839377256030943803152700737297124384897921408"
},
"0,5": {
"1": "2.496599250749765356184001781151499743240611432798116223272910138242101414127046304503946306551384849",
"3": "0.07367402685456878284757580216399993251860559423656368198422467751502329299786315068530571003521550193",
"5": "0.01893533579656526200673898655484419383093983469259244266748142803448195949955683298192357740459948566",
"7": "0.01459550287778226738933132941889038213953759292632592657627018777626197412053554396231447538121862766"
},
"1,4": {
"1": "1.071285055421807658518711978030817160763179777167056217024693659950183871493064087996272300093743097",
"3": "0.08593729069176845242384174578764695803378737791306498064316846696375790752297230255565160098338193125",
"5": "0.05451425313276188036303391980200959687761434954457591364994026463408579936330354610551573828247090613",
"7": "0.1056198399566258441435137931548286612812846899361471170048748828753902794532917354178521540772430069"
},
"2,3": {
"1": "0.6603448690186723578372668317059942638542419916968738583008035875538949586837994454109810720121753277",
"3": "0.2522539448978419659945096125550904087750684507559700999206593094528971020741986059081563549596517412",
"5": "1.043297367386871344918931607894771221756616331226915578868832558986627109643922022677421150635684261",
"7": "16.27973600660073218435050503817333790350292233284149984276094133768055181712231644550825332870633761"
},
"0,6": {
"1": "7.297134851598829596710435738816456522509940731668314950669322480828795045115062766414283077394726373",
"3": "0.09474141900486840158430585401885412049016015774898307820079945392873927402098402684560761206826320372",
"5": "0.01331598489189104085062675158291232217560738812954071869068484197618400279935439929233445654184610513",
"7": "0.006067414482173076761085528683928195506250602689193735262096162623220855170050295164238479276383462215"
},
"1,5": {
"1": "2.515320118870661382904988671360713183760828845747051998301714842527010245232359016262991349469318363",
"3": "0.07864644417421688937874504705648889434879134905592020675997649610054155575425039361623637622891488114",
"5": "0.02297649329259878932476877301500180072396676358693213302770239560405571873120229756741818445165455794",
"7": "0.02151723418850830021520886507013523862661720717601510097001490942795358516116056748901009997742580604"
},
"2,4": {
"1": "1.093670186350183889271856823917256856644045856785500562365575315163594611211087327750765401655813100",
"3": "0.1014289491858825870406042591591014600057995675551832080733171830658094210472055830394048130010672143",
"5": "0.08936278205693961277649511639845986648895654900860516057640013792036960509153194309510942085804287522",
"7": "0.2835956106065839229529521396178372227886931966440129676861149006501250412292385861744117577409068171"
}
},
"Keywords": [
"IKM",
"Bessel moments",
"vacuum integrals",
"sunrise integrals",
"Ising-type integrals"
],
"Comments": {
"functions": "Point values of HREF{Values_of_modified_Bessel_functions_of_the_first_kind}[$I_\\nu(x)$] and HREF{Values_of_modified_Bessel_functions_of_the_second_kind}[$K_\\nu(x)$] are tabulated separately. The integral families and their applications are developed in CITE{BBBG}; the full text is available in CITE{BBBGtext}.",
"normalization": "Every Bessel argument is $t$, with a common unit scale. The values contain no additional powers of $\\pi$, factorials or loop-measure factors. The integrals converge ordinarily: the integrand is $O(t^k(1+|\\log t|)^b)$ at zero and $O(t^{k-(a+b)/2}e^{-(b-a)t})$ at infinity.",
"notation": "In CITE{BBBG}, the cases $a=0,1,2$ are denoted $c_{a+b,k}$, $s_{a+b,k}$ and $t_{a+b,k}$, respectively. These are exponent specializations of the same integral.",
"application": "The moments with $a=0$ and $k=1$ are equal-mass vacuum integrals in two spacetime dimensions; the moments with $a=1$ and $k=1$ include on-shell sunrise integrals. Applying the mass differential operator $\\partial_m^2+m^{-1}\\partial_m$ to a factor $K_0(mt)$ multiplies that factor by $t^2$, producing the higher odd moment powers CITE{BBBG}. The two-$I_0$ moments with five factors occur in lattice Green-function evaluations in Section 5 of that reference."
},
"Formulas": {
"closed": "$\\operatorname{IKM}(1,2;1)=\\pi/(3\\sqrt3)$, $\\operatorname{IKM}(1,3;1)=\\pi^2/16$, $\\operatorname{IKM}(0,3;1)=\\tfrac34L(2,\\chi_{-3})$, $\\operatorname{IKM}(0,4;1)=\\tfrac78\\zeta(3)$ and $\\operatorname{IKM}(0,4;3)=\\tfrac7{32}\\zeta(3)-\\tfrac3{16}$ CITE{BBBG}. Here $\\chi_{-3}(n)$ is $0,1,-1$ for $n\\equiv0,1,2\\pmod3$, respectively. The HREF{Values_of_Dirichlet_L-functions_at_positive_integers#3,2,2}[$L$-value $L(2,\\chi_{-3})$] and HREF{Values_of_the_Riemann_zeta_function_at_rational_numbers#3}[$\\zeta(3)$] are tabulated separately.",
"combinatorial": "For $r\\geq0$, let $A_r=\\sum_{j_1+j_2+j_3=r}\\binom{r}{j_1,j_2,j_3}^{\\!2}$ and $B_r=\\sum_{j_1+j_2+j_3+j_4=r}\\binom{r}{j_1,j_2,j_3,j_4}^{\\!2}$, where all $j_i$ are nonnegative integers. Then $\\operatorname{IKM}(1,2;2r+1)=\\frac{\\pi}{3\\sqrt3}\\left(\\frac{2^r r!}{3^r}\\right)^2 A_r$ and $\\operatorname{IKM}(1,3;2r+1)=\\frac{\\pi^2}{16}\\left(\\frac{r!}{4^r}\\right)^2 B_r$ CITE{BBBG}.",
"recurrence-three": "For $a+b=3$ and $M_k=\\operatorname{IKM}(a,b;k)$, $(k+1)^4M_k-2(5k^2+20k+21)M_{k+2}+9M_{k+4}=0$ CITE{BBBG}.",
"recurrence-four": "For $a+b=4$ and $M_k=\\operatorname{IKM}(a,b;k)$, $(k+1)^5M_k-4(k+2)(5k^2+20k+23)M_{k+2}+64(k+3)M_{k+4}=0$ CITE{BBBG}.",
"recurrence-five": "For $a+b=5$ and $M_k=\\operatorname{IKM}(a,b;k)$, $(k+1)^6M_k-[35(k+2)^4+42(k+2)^2+3]M_{k+2}+[259(k+3)^2+104]M_{k+4}-225M_{k+6}=0$ CITE{BBBG}.",
"recurrence-six": "For $a+b=6$ and $M_k=\\operatorname{IKM}(a,b;k)$, $(k+1)^7M_k-(k+2)[56(k+2)^4+112(k+2)^2+24]M_{k+2}+(k+3)[784(k+3)^2+944]M_{k+4}-2304(k+4)M_{k+6}=0$ CITE{BBBG}."
},
"References": {
"BBBG": {
"bib": "D. H. Bailey, J. M. Borwein, D. Broadhurst and M. L. Glasser, Elliptic integral evaluations of Bessel moments and applications, Journal of Physics A: Mathematical and Theoretical 41 (2008), 205203. See Sections 1 and 3–6; equations (8)–(13), (23)–(24) and (55)–(59) in the arXiv version.",
"arxiv": "0801.0891",
"doi": "10.1088/1751-8113/41/20/205203"
},
"Zhou": {
"bib": "Y. Zhou, Hilbert Transforms and Sum Rules of Bessel Moments, Ramanujan Journal 48 (2019), 159–172. Equation (1.2) defines IKM.",
"arxiv": "1706.01068",
"doi": "10.1007/s11139-017-9945-y"
}
},
"Similar tables": [
{
"table": "HREF{Values_of_modified_Bessel_functions_of_the_first_kind}[$I_\\nu(x)$]",
"relation": "point values of the first-kind factor in the integrand"
},
{
"table": "HREF{Values_of_modified_Bessel_functions_of_the_second_kind}[$K_\\nu(x)$]",
"relation": "point values of the second-kind factor in the integrand"
},
{
"table": "HREF{Values_of_Dirichlet_L-functions_at_positive_integers}[Dirichlet L-values]",
"relation": "contains $L(1,\\chi_{-3})=\\operatorname{IKM}(1,2;1)$ and $L(2,\\chi_{-3})=\\tfrac43\\operatorname{IKM}(0,3;1)$"
},
{
"table": "HREF{Values_of_the_Riemann_zeta_function_at_rational_numbers}[Riemann zeta values]",
"relation": "$\\zeta(3)$ occurs in the four-$K_0$ moments"
},
{
"table": "HREF{Watson_integrals_of_the_cubic_lattices}[Watson integrals of the cubic lattices]",
"relation": "lattice Green functions related to five-factor Bessel moments in Section 5 of CITE{BBBG}"
},
{
"table": "HREF{Feynman_periods_of_the_primitive_log-divergent_graphs}[Primitive phi-four graph periods]",
"relation": "massless primitive graph periods, a different family of Feynman integrals"
},
{
"table": "HREF{Feynman_periods_of_the_primitive_log-divergent_non-_phi_4_graphs}[Primitive non-phi-four graph periods]",
"relation": "massless primitive graph periods for graphs outside the phi-four family"
}
],
"Programs": {
"program-mpmath": {
"language": "Python",
"code": "from mpmath import mp\nmp.dps = 150\ndef IKM(a, b, k):\n if not (0 <= a < b and k >= 0):\n raise ValueError(\"Require 0 <= a < b and k >= 0\")\n return mp.quad(lambda t: t**k * mp.besseli(0,t)**a\n * mp.besselk(0,t)**b, [0, 1, 4, 16, mp.inf])\nprint(IKM(0, 3, 1))"
}
}
}