Bessel moments

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Numbers

$a$, $b$

$k$

$\operatorname{IKM}(a,b;k)$

0, 3

6.948822781079629789436436445470829757674851132609891735162380688191422338199804186839952351806095537

0, 3

0.5859768096723647226503905722180692672738507524089640651665750472251652384887136635696521781241573958

0, 3

0.1880051289185344914779605663063667920623719000573058401281020442919023938867790139257798139211350246

0, 3

0.1146357462298196302005207629574256896984676698786187535554333963002203179849515514262029041655431944

0, 3

0.1300955220428411573379623373993394639863792878099444760482968560019583820884582348995813703603839498

0, 3

0.3819245022180162575370215230591742624177618932421785615533175844041234313493197810651735404930775110

1, 2

0.6045997880780726168646927525473852440946887493642468585232949784627077270421179612280416627373533896

1, 2

0.8061330507707634891529236700631803254595849991523291446977266379502769693894906149707222169831378528

1, 2

7.165627117962342125803765956117158448529644436909592397313125670669128616795472133073086373183447581

1, 2

177.7075525254660847199333957117055295235351820353578914533655166325943896965277089002125420549495000

0, 4

27.24133841780597340670998026455793502399788809861827465512290187919531478484839302736940746053615985

0, 4

1.051799790264644999724770891322518741919363005797936521568237610924108430063023953913148889408168351

0, 4

0.1957706252472879172174580832755772374188278969665250281979338461663529929694444626553529111638580858

0, 4

0.07544994756616124993119272283062968547984075144948413039205940273102710751575598847828722235204208772

0, 4

0.03828740824078218737958726495360194958972131503659722818610395477929743815257297983700263911607365351

0, 4

0.05404952809545124938073450547566716931856676304535717352853462457924396764180389630458500116837878948

1, 3

6.997563016680632359556757826853096005697754284353362908336255807271368862538584419940117092896528787

1, 3

0.6168502750680849136771556874922594459571062129525494141508343360137528014012003276876108377324095145

1, 3

0.1542125687670212284192889218730648614892765532381373535377085840034382003503000819219027094331023786

1, 3

0.2698719953422871497337556132778635076062339681667403686909900220060168506130251433633297415079291626

1, 3

1.387913118903191055773600296857583753403488979143236181839377256030943803152700737297124384897921408

0, 5

2.496599250749765356184001781151499743240611432798116223272910138242101414127046304503946306551384849

0, 5

0.07367402685456878284757580216399993251860559423656368198422467751502329299786315068530571003521550193

0, 5

0.01893533579656526200673898655484419383093983469259244266748142803448195949955683298192357740459948566

0, 5

0.01459550287778226738933132941889038213953759292632592657627018777626197412053554396231447538121862766

1, 4

1.071285055421807658518711978030817160763179777167056217024693659950183871493064087996272300093743097

1, 4

0.08593729069176845242384174578764695803378737791306498064316846696375790752297230255565160098338193125

1, 4

0.05451425313276188036303391980200959687761434954457591364994026463408579936330354610551573828247090613

1, 4

0.1056198399566258441435137931548286612812846899361471170048748828753902794532917354178521540772430069

2, 3

0.6603448690186723578372668317059942638542419916968738583008035875538949586837994454109810720121753277

2, 3

0.2522539448978419659945096125550904087750684507559700999206593094528971020741986059081563549596517412

2, 3

1.043297367386871344918931607894771221756616331226915578868832558986627109643922022677421150635684261

2, 3

16.27973600660073218435050503817333790350292233284149984276094133768055181712231644550825332870633761

0, 6

7.297134851598829596710435738816456522509940731668314950669322480828795045115062766414283077394726373

0, 6

0.09474141900486840158430585401885412049016015774898307820079945392873927402098402684560761206826320372

0, 6

0.01331598489189104085062675158291232217560738812954071869068484197618400279935439929233445654184610513

0, 6

0.006067414482173076761085528683928195506250602689193735262096162623220855170050295164238479276383462215

1, 5

2.515320118870661382904988671360713183760828845747051998301714842527010245232359016262991349469318363

1, 5

0.07864644417421688937874504705648889434879134905592020675997649610054155575425039361623637622891488114

1, 5

0.02297649329259878932476877301500180072396676358693213302770239560405571873120229756741818445165455794

1, 5

0.02151723418850830021520886507013523862661720717601510097001490942795358516116056748901009997742580604

2, 4

1.093670186350183889271856823917256856644045856785500562365575315163594611211087327750765401655813100

2, 4

0.1014289491858825870406042591591014600057995675551832080733171830658094210472055830394048130010672143

2, 4

0.08936278205693961277649511639845986648895654900860516057640013792036960509153194309510942085804287522

2, 4

0.2835956106065839229529521396178372227886931966440129676861149006501250412292385861744117577409068171

Definition

The Bessel moments [2] 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

$a$

— number of first-kind factors ($a\geq0$)

$b$

— number of second-kind factors ($b>a$)

$k$

— moment power ($k\geq0$)

Formulas

(1)

$\operatorname{IKM}(1,2;1)=\pi/(3\sqrt3)$, $\operatorname{IKM}(1,3;1)=\pi^2/16$, $\operatorname{IKM}(0,3;1)=\tfrac34L(2,\chi_{-3})$, $\operatorname{IKM}(0,4;1)=\tfrac78\zeta(3)$ and $\operatorname{IKM}(0,4;3)=\tfrac7{32}\zeta(3)-\tfrac3{16}$ [1]. Here $\chi_{-3}(n)$ is $0,1,-1$ for $n\equiv0,1,2\pmod3$, respectively. The $L$-value $L(2,\chi_{-3})$ and $\zeta(3)$ are tabulated separately.

(2)

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 $\operatorname{IKM}(1,3;2r+1)=\frac{\pi^2}{16}\left(\frac{r!}{4^r}\right)^2 B_r$ [1].

(3)

For $a+b=3$ and $M_k=\operatorname{IKM}(a,b;k)$, $(k+1)^4M_k-2(5k^2+20k+21)M_{k+2}+9M_{k+4}=0$ [1].

(4)

For $a+b=4$ and $M_k=\operatorname{IKM}(a,b;k)$, $(k+1)^5M_k-4(k+2)(5k^2+20k+23)M_{k+2}+64(k+3)M_{k+4}=0$ [1].

(5)

For $a+b=5$ and $M_k=\operatorname{IKM}(a,b;k)$, $(k+1)^6M_k-[35(k+2)^4+42(k+2)^2+3]M_{k+2}+[259(k+3)^2+104]M_{k+4}-225M_{k+6}=0$ [1].

(6)

For $a+b=6$ and $M_k=\operatorname{IKM}(a,b;k)$, $(k+1)^7M_k-(k+2)[56(k+2)^4+112(k+2)^2+24]M_{k+2}+(k+3)[784(k+3)^2+944]M_{k+4}-2304(k+4)M_{k+6}=0$ [1].

Comments

(7)

Point values of $I_\nu(x)$ and $K_\nu(x)$ are tabulated separately. The integral families and their applications are developed in [1]; the full text is available in [3].

(8)

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.

(9)

In [1], 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.

(10)

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 [1]. The two-$I_0$ moments with five factors occur in lattice Green-function evaluations in Section 5 of that reference.

(11)

The added $k=0,2$ three- and four-$K_0$ moments are initial values for the even recurrences. Their gamma and elliptic-integral evaluations, and the complementary elliptic-integral representation of $\operatorname{IKM}(1,3;0)$, are given in Sections 3.3, 4.4, 4.6 and 4.11 of [1]. The two-dimensional physical interpretation of the odd moments is not asserted for these even moments.

Programs

(P1)

Python

from mpmath import mp
mp.dps = 150
def IKM(a, b, k):
    if not (0 <= a < b and k >= 0):
        raise ValueError("Require 0 <= a < b and k >= 0")
    return mp.quad(lambda t: t**k * mp.besseli(0,t)**a
                   * mp.besselk(0,t)**b, [0, 1, 4, 16, mp.inf])
print(IKM(0, 3, 1))

References

[1]

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) (doi)

[2]

Y. Zhou, Hilbert Transforms and Sum Rules of Bessel Moments, Ramanujan Journal 48 (2019), 159–172. Equation (1.2) defines IKM. (arXiv) (doi)

Links

[3]

Full text: Elliptic integral evaluations of Bessel moments

Similar tables

$I_\nu(x)$ — point values of the first-kind factor in the integrand

$K_\nu(x)$ — point values of the second-kind factor in the integrand

Dirichlet L-values — contains $L(1,\chi_{-3})=\operatorname{IKM}(1,2;1)$ and $L(2,\chi_{-3})=\tfrac43\operatorname{IKM}(0,3;1)$

Riemann zeta values — $\zeta(3)$ occurs in the four-$K_0$ moments

Watson integrals of the cubic lattices — lattice Green functions related to five-factor Bessel moments in Section 5 of [1]

Primitive phi-four graph periods — massless primitive graph periods, a different family of Feynman integrals

Primitive non-phi-four graph periods — massless primitive graph periods for graphs outside the phi-four family

Data properties

Entries are of type: real number

How well the digits are known: heuristic (agreement-checked)

digits: 100 (Unknown key)

Table is complete: no (it preserves all 40 moments with $3\leq a+b\leq6$, $0\leq a<b$ and $k=1,3,5,7$, and adds $(a,b,k)=(0,3,0),(0,3,2),(0,4,0),(0,4,2),(1,3,0)$. The odd moments form short low-factor vacuum, sunrise and lattice-integral prefixes. The five even moments are the initial three- and four-$K_0$ even-recurrence values and the mixed zeroth moment studied in Sections 3.3, 4.4, 4.6 and 4.11 of [1]; this is a bounded selection, not all even moments)

How they were obtained:

Original 40 odd moments, retained unchanged:

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.

The 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.

After 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.

Five even additions: the same Arb-Bessel/tanh-sinh method, ported from the Sage Arb interface to python-flint, runs at 150 and 200 working digits. For k=0 the substitution t=u^2 makes the transformed integrand tend to zero at the lower endpoint. The unchanged upper-tail estimate applies and is below 4e-254 and 2e-318 for the two runs. The largest relative 150/200-digit discrepancy is below 4.1e-152. Independent gamma and elliptic-integral calculations at 140 and 180 digits check all five additions, with relative differences below 2.2e-180 from the Bessel quadrature. These comparisons are not certified integration enclosures; the rigour label remains heuristic (agreement-checked). All 40 original values are reproduced within their stored last-place intervals, without altering their decimal strings. The nonzero controls IKM(0,1;0)=pi/2, IKM(0,2;0)=pi^2/4, IKM(0,1;1)=1 and IKM(0,2;1)=1/2 pass before the retained-row reproduction. The current default generator enumerates all 45 rows; even-checks.py and even-checks-evidence.json record the extension checks. The older checks.py describes the original odd slice.