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Laguerre–Selberg integrals for classical Wishart ensembles
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Numbers
$\beta$
$n$
$r$ 
$Z_{\beta,n}(r)$
1
1
0:
1.772453850905516027298167483341145182797549456122387128213807789852911284591032181374950656738544665
1
1
1:
1
1
1
2:
0.8862269254527580136490837416705725913987747280611935641069038949264556422955160906874753283692723327
1
1
3:
1
1
1
4:
1.329340388179137020473625612505858887098162092091790346160355842389683463443274136031212992553908499
1
1
5:
2
1
1
6:
3.323350970447842551184064031264647217745405230229475865400889605974208658608185340078032481384771248
1
2
0:
2
1
2
1:
1
1
2
2:
1
1
2
3:
3/2
1
2
4:
3
1
2
5:
15/2
1
2
6:
45/2
1
3
0:
2.658680776358274040947251225011717774196324184183580692320711684779366926886548272062425985107816998
1
3
1:
3/2
1
3
2:
1.994010582268705530710438418758788330647243138137685519240533763584525195164911204046819488830862749
1
3
3:
9/2
1
3
4:
14.95507936701529148032828814069091247985432353603264139430400322688393896373683403035114616623147061
1
3
5:
135/2
1
3
6:
392.5708333841514013586175636931364525961759928208568366004800847057033977980918932967175868635761036
1
4
0:
6
1
4
1:
9/2
1
4
2:
9
1
4
3:
135/4
1
4
4:
405/2
1
4
5:
14175/8
1
4
6:
42525/2
1
4
8:
13395375/2
1
5
0:
29.91015873403058296065657628138182495970864707206528278860800645376787792747366806070229233246294123
1
5
1:
135/4
1
5
2:
112.1630952526146861024621610551818435989074265202448104572800242016295422280262552276335962467360296
1
5
3:
6075/8
1
5
4:
8832.843751143406530568895183095570183413959838469278823510801905878326450457067599176145704430462332
1
5
5:
637875/4
1
5
6:
4173518.672415259585693802974012656911663096023676734244108853900527509247840964440610728845343393452
1
5
10:
34896785083679.23394197529651526677424000103477174749545355459200808169453655742764906947088064529246
1
6
0:
405
1
6
1:
6075/8
1
6
2:
18225/4
1
6
3:
1913625/32
1
6
4:
5740875/4
1
6
5:
1808375625/32
1
6
6:
27125634375/8
1
6
12:
1716401298919334762109375/16
1
7
0:
17665.68750228681306113779036619114036682791967693855764702160381175665290091413519835229140886092466
1
7
1:
1913625/32
1
7
2:
695586.4454025432642823004956687761519438493372794557073514756500879182079734940734351214742238989086
1
7
7:
88820855338359375/32
1
7
14:
333241325466264963683295607544599043.1255512392294330565552166481545779290825413523596319296604963782
1
8
0:
5740875/2
1
8
1:
602791875/32
1
8
2:
1808375625/4
1
8
8:
3780883980136996154296875/32
1
8
16:
52778426763472994229677318535382902936350555419921875/32
1
9
0:
1971987572.716210154240321905220980390760812871187256930341433467999248119604855698188569379424753406
1
9
1:
1708914965625/64
1
9
2:
1397646192162.613946817828150325369851951726122453968349379490970444467104769941476091148547667293976
1
9
9:
21727287901788446775096371337890625/64
1
9
18:
19301815436454335366976379762617879882890670924819920988518334000452691.57775466481993936325041057757
1
10
0:
25633724484375/4
1
10
1:
24223869637734375/128
1
10
2:
363358044566015625/16
1
10
10:
10612962094054598975995715785743466644287109375/128
1
10
20:
775022984539723345394185560533953773302557497857384862170309684125067352137826383113861083984375/1024
1
11
0:
108963991256477789.8287849271697416470827864478218175074384985647832767666556265623297561686475114066
1
11
1:
3777106873263732421875/512
1
11
2:
2123776292083287422381.661221117745790172934609577111855918486089229053730097321466408403824545402010
1
12
0:
169969809296867958984375/16
1
12
1:
1766836167640942433642578125/1024
1
12
2:
79507627543842409513916015625/64
1
13
0:
6457421457690188531029780255.104297990487939165467067739614753897529283955875784568169742143673715264
1
13
1:
10744263248137144009663040771484375/4096
1
13
2:
4908508023852920402666490120601046.112813305582580331119334723694677236303997167080360351838292579756
1
14
0:
3384442923163200363043857843017578125/128
1
14
1:
457356694421659081059931729616180419921875/16384
1
14
2:
144067358742822610533878494829096832275390625/1024
1
15
0:
783536392567724195863184568266017670374999.5214467292853645956284047460873799012860840610720699325478
1
15
1:
292028137855670002117435055980950006443023681640625/131072
1
15
2:
31268629667846638031729067833698044219496454535160.95470905271111907163049362137421210820792518467701
1
16
0:
91988863424536050666992042633999252029552459716796875/512
1
16
1:
186463725883120188103259545220182333845203574657440185546875/131072
1
16
2:
58736073653182859252526756744357435161239126017093658447265625/1024
2
1
0:
1
2
1
1:
1
2
1
2:
2
2
1
3:
6
2
1
4:
24
2
1
5:
120
2
1
6:
720
2
2
0:
2
2
2
1:
4
2
2
2:
24
2
2
3:
288
2
2
4:
5760
2
2
5:
172800
2
2
6:
7257600
2
3
0:
24
2
3
1:
144
2
3
2:
3456
2
3
3:
207360
2
3
4:
24883200
2
3
5:
5225472000
2
3
6:
1755758592000
2
4
0:
3456
2
4
1:
82944
2
4
2:
9953280
2
4
3:
3583180800
2
4
4:
3009871872000
2
4
5:
5056584744960000
2
4
6:
15291112268759040000
2
4
8:
610372270209600847872000000
2
5
0:
9953280
2
5
1:
1194393600
2
5
2:
859963392000
2
5
3:
2167107747840000
2
5
4:
14562964065484800000
2
5
5:
220192016670130176000000
2
5
6:
6658606584104736522240000000
2
5
10:
1301721617754709336555873881119155814400000000000
2
6
0:
859963392000
2
6
1:
619173642240000
2
6
2:
3120635156889600000
2
6
3:
62912004762894336000000
2
6
4:
3804918048059849441280000000
2
6
5:
575303608866649235521536000000000
2
6
6:
191368992453402201703883735040000000000
2
6
12:
62967954103709550461572791449149849215073626500479427857612800000000000000000
2
7
0:
3120635156889600000
2
7
1:
15728001190723584000000
2
7
2:
634153008009974906880000000
2
7
7:
3995644955882815849861180515460392302306918400000000000000
2
7
14:
201601192717590850422720672730323387417870409205771511413350155133633857365134782995890176000000000000000000000000
2
8
0:
634153008009974906880000000
2
8
1:
25569049282962188245401600000000
2
8
2:
9278496603801318870491332608000000000
2
8
8:
3644054310450804456043334309647191289028137348538101854130667520000000000000000000
2
8
16:
105854808278437423342986357948945186726490143416543730801216752281562222759570411368449453181653089940824587140806367287810457600000000000000000000000000000000
2
9
0:
9278496603801318870491332608000000000
2
9
1:
3366980847587422591723894776791040000000000
2
9
2:
12218100099725239100847669366019325952000000000000
2
9
9:
244070594721506898495680894109254877168612712156429897264564594575831583038485708144640000000000000000000000000
2
9
18:
20033456605923643838449560348243503757332054427734153490180323115736604371644144248888332437237744329448440047017502849907576467468911161128722514816378797951630138408960000000000000000000000000000000000000000000
2
10
0:
12218100099725239100847669366019325952000000000000
2
10
1:
44337041641882947649156022595410930014617600000000000000
2
10
2:
1769792823810713244721831122736499011207487815680000000000000000
2
10
10:
1900864241152215153881159781849768786183581652886944594495487680672962801214870693414067367488017494989146056294400000000000000000000000000000000
2
10
20:
2739969411912212580760954411026929090086532429921069973194801486343003037400670764682303447557698144305439384045766929825404848024677999276575916323301740915919231255181874237618090293795386098909922478521803211800576000000000000000000000000000000000000000000000000000000000
2
11
0:
1769792823810713244721831122736499011207487815680000000000000000
2
11
1:
70644466189487478446912388560048283730567049640935424000000000000000000
2
11
2:
33838812335910405356036549180084823984195585685287493592678400000000000000000000
2
12
0:
33838812335910405356036549180084823984195585685287493592678400000000000000000000
2
12
1:
16208845251000821622190076715739318824148060256189805890882701885440000000000000000000000
2
12
2:
100932816521963337058467349262504425695801513494947250030489115000834097152000000000000000000000000
2
13
0:
100932816521963337058467349262504425695801513494947250030489115000834097152000000000000000000000000
2
13
1:
628510747884849356700486999978479718899810497204517220842656353283785940314724761600000000000000000000000000
2
13
2:
54792493001435181286567726865938298667541823150112188234035804948103966943222894914615377920000000000000000000000000000
2
14
0:
54792493001435181286567726865938298667541823150112188234035804948103966943222894914615377920000000000000000000000000000
2
14
1:
4776715910453078252125191941240832362471533046759381658535987154992208518051459399373338552307810304000000000000000000000000000000
2
14
2:
6246388959317273697002355028640798595388908894912188858396838808160554318767158761553190087439150780747087872000000000000000000000000000000000
2
15
0:
6246388959317273697002355028640798595388908894912188858396838808160554318767158761553190087439150780747087872000000000000000000000000000000000
2
15
1:
8168242734657393593210578106589478202240479153363874980900727681659226111323514872669730545975856235670154700858064896000000000000000000000000000000000000
2
15
2:
170902426491419181816462269063184520697033516174276484634885938269661339143905199274511845831028363939821257678508785129276571648000000000000000000000000000000000000000
2
16
0:
170902426491419181816462269063184520697033516174276484634885938269661339143905199274511845831028363939821257678508785129276571648000000000000000000000000000000000000000
2
16
1:
3575755560829328576078710235088732322718019563808236478434978881261862483824638280211381384089655209681276050683345964421992626032081895424000000000000000000000000000000000000000000
2
16
2:
1271851298931353962031954312379542734216556650885563396355905805440107233962415991365894815734222195810940678354959469385039276372828346967807391432704000000000000000000000000000000000000000000000
4
1
0:
1
4
1
1:
6
4
1
2:
120
4
1
3:
5040
4
1
4:
362880
4
1
5:
39916800
4
1
6:
6227020800
4
2
0:
72
4
2
1:
8640
4
2
2:
7257600
4
2
3:
21946982400
4
2
4:
173820100608000
4
2
5:
2982752926433280000
4
2
6:
97714985869954252800000
4
3
0:
3110400
4
3
1:
15676416000
4
3
2:
948109639680000
4
3
3:
315379190543155200000
4
3
4:
389657297499879112704000000
4
3
5:
1404169037270564370540134400000000
4
3
6:
12512157123787563347859386400768000000000
4
4
0:
316036546560000
4
4
1:
114683342015692800000
4
4
2:
762965337762001059840000000
4
4
3:
39591675232691717077047705600000000
4
4
4:
10272424382534006759437136056811520000000000
4
4
5:
10068816713332676718259684470456670224384000000000000
4
4
6:
30684379010629090187730379976768979891623464796160000000000000
4
4
8:
4977092944386925548172385777336709429477650494841207072558929674240000000000000000
4
5
0:
208081455753273016320000000
4
5
1:
8305945853012248337842176000000000
4
5
2:
8620216265063502175751442844876800000000000
4
5
3:
93936965470335297396186607893853262315520000000000000
4
5
4:
6629404295890143038416051611079975841314790965248000000000000000
4
5
5:
2222317436133979982417833439475021994806466494717532255027200000000000000000
4
5
6:
2844423691696297247566739211912149398020668030063983510988714158325760000000000000000000
4
5
10:
311682875480281002953261128485077594025854110268657070154810938044411882820101216533293980717358543851137007616000000000000000000000000000000
4
6
0:
1989280676553115886711871425740800000000000
4
6
1:
12387292149934324931365266975013617008640000000000000
4
6
2:
2699757405566121599354953144783700568861227089920000000000000000
4
6
3:
8002248067241194887867978472222513112996850059568727719936000000000000000000
4
6
4:
193141720165860507555662939771467500920136429800391278555156508888268800000000000000000000
4
6
5:
27192990674433603896421825115498764223181353417699993040652125270400570422093414400000000000000000000000
4
6
6:
17611473116477287711403985937593422587215949786812995275364531056968695480283110428055300997120000000000000000000000000
4
6
12:
41698619221649872152184841453365107495813423400572640007804923862380412689700971256417502465166584839268669447601391057101484603072139487400081078649331705006330709110947840000000000000000000000000000000000000000000000
4
7
0:
539951481113224319870990628956740113772245417984000000000000000
4
7
1:
706080711815399548929527512254927627617369122903123034112000000000000000000
4
7
2:
41857338735635404114230358464404721561639473950549193494925559201792000000000000000000000
4
7
7:
20821988102119826870896392768252564138327448248094685058783636416289060221668313630118142472404567250021537552878567841837754100041646080000000000000000000000000000000000000
4
7
14:
101070039582597846251690305275032374622488864333337597538722743789953089776213481797112788128301185103153068770605287135462379687625576112052933952779444868460619578626738288027773145390406458065143011599138116961216581727545722585291024767292028097658880000000000000000000000000000000000000000000000000000000000000
4
8
0:
7386589188641541902511239729012597922642260108920445910869216329728000000000000000000000
4
8
1:
2627316910909629415368037622944986948687977862822570233106551608139997027237888000000000000000000000000
4
8
2:
53266704905570698086065347765108464889281473354473533979457802428128076572857812364753371136000000000000000000000000000
4
8
8:
12110798264641870849043293271360451606674100177865862116653269085838826857111549185201006553029503329937860266546792819032257329999791382707365579094811570993319428570143763303522695852851200000000000000000000000000000000000000000000000000
4
8
16:
27181757124889060069122894903729659298248513688818670533344246265153911968602321026495669778100283416758639830176538962108869657289208755127603152092448633499720946853068585116936787026581250615647536980800178601332870790812991626491451799299422909174512018703819329285097935194178689904815115893422139862544346266052290143920360644805324521770424729600000000000000000000000000000000000000000000000000000000000000000000000000000000
4
9
0:
8410532353511162855694528594490810245676022108601084312545968804441275248345970373382111232000000000000000000000000000
4
9
1:
1023100052634595519847922232948999575759151211363209232011684131479932889969905842998884018718318499201024000000000000000000000000000000
4
9
2:
8711857604171167333311160827283906344392944370471482197511965185886330186441748157405206436521803376546425127342713077760000000000000000000000000000000000
4
9
9:
9779881997402895947530437371285286131641722470243424388138930803033796198152870147078425135405840654965472074659079648717862949440334805953376630268383896671323552701849483694193151409689184950624145559579758851940323775074682446363958504367646887116800000000000000000000000000000000000000000000000000000000000000000
4
9
18:
3911509554650712745523045044658796338351466577376898915295551815439953180160250234208326806411498347340779970085603249480933630961933441634195037049365284708480442194965863682765109271060071550844745111791113318593409121891166299240655944355144106719914161402512345425818848585541595171253911493226879792597958736094760678621526346829536543959064337628387577154784994287986526682166827477980931547787592373819578643556723847443818193157556325938282400153304200904704000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
4
10
0:
1244551086310166761901594403897700906341849195781640313930280740840904312348821165343600919503114768078060732477530439680000000000000000000000000000000000
4
10
1:
63585287580410894124352570389798121139666889349877803915989585960412378502340314493943491610777426906584567437900708656631942940406579200000000000000000000000000000000000000
4
10
2:
273967986479266241662775711005324915343757848456988383257825227567070320080521683701836751400895672189528299144788950023392876900236145430632564049051648000000000000000000000000000000000000000000
4
10
10:
27457156853387949649664308424054091551486006652809555361890993158670204395599733640520649886729378185769623678733785293805172090410382960534924820553868313821516247189105902998343316917739162412381541353283686135587257320200926558620281192634315140296356189619220367507732350838684763102183251686938289963733955876801489141760000000000000000000000000000000000000000000000000000000000000000000000000000000000
4
10
20:
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Definition
For $c=\beta/2$ and $a=c(r+1)$, the Laguerre–Selberg integral [3] is $Z_{\beta,n}(r)=\int_{[0,\infty)^n}\prod_{i=1}^n x_i^{a-1}e^{-x_i}\prod_{i<j}|x_i-x_j|^{\beta}\,dx_1\cdots dx_n$.
Parameters
$\beta$
—   Dyson exponent ($\beta\in\{1,2,4\}$)
$n$
—   number of eigenvalues ($n\geq1$)
$r$
—   excess column count ($r\geq0$)
Formulas
(1)
$Z_{\beta,n}(r)=\prod_{j=0}^{n-1}\frac{\Gamma(a+jc)\Gamma(1+(j+1)c)}{\Gamma(1+c)}$, where $c=\beta/2$ and $a=c(r+1)$ [3].
(2)
$Z_{\beta,1}(r)=\Gamma\!\left(\frac{\beta(r+1)}2\right)$.
(3)
$Z_{\beta,n}(r+2)=Z_{\beta,n}(r)\prod_{j=0}^{n-1}\prod_{k=0}^{\beta-1}(a+jc+k)$, where $c=\beta/2$ and $a=c(r+1)$.
Comments
(4)
The integral uses labelled coordinates on the full positive orthant. The integral over a single ordered chamber is $Z_{\beta,n}(r)/n!$. The exponential weight is $e^{-x}$; the eigenvalues $\lambda$ in [1], equation (3), are related by $\lambda=2x$.
(5)
The integral normalizes the rescaled eigenvalues of a classical Wishart ensemble with $n$ rows and $n+r$ columns [1]. The Dyson exponents $\beta=1,2,4$ correspond to real, complex and quaternion matrices, respectively. The number $n$ counts each quaternion eigenvalue once, rather than twice in its complex representation. The excess column count is $r=m-n$ for an $n\times m$ matrix.
(6)
When $n=1$, the integral is a value of the Gamma function, as stated in (2).
(7)
All Gamma arguments in (1) are positive integers or half-integers. Their reduction gives a positive rational multiple of $\sqrt{\pi}$ precisely when $\beta=1$, $n$ is odd and $r$ is even; all other values are rational. Rational values are stored exactly.
Programs
(P1)
Sage
import numberdb.sage as numberdb
from sage.rings.rational_field import QQ
from sage.rings.real_arb import RealBallField


def laguerre_integral(beta, n, r, digits=100):
    if beta not in (1, 2, 4) or n < 1 or r < 0:
        raise ValueError('Require beta in {1,2,4}, n >= 1, r >= 0')
    R = RealBallField(numberdb.bits(digits, losing=96))
    c = QQ(beta) / QQ(2)
    a = c*(r+1)
    z = R(1)
    for j in range(n):
        z *= R(a+j*c).gamma()*R(1+(j+1)*c).gamma()/R(1+c).gamma()
    return z


if __name__ == '__main__':
    print(laguerre_integral(1, 3, 0))
References
[1]
Ioana Dumitriu and Alan Edelman, Matrix Models for Beta Ensembles, Journal of Mathematical Physics 43 (2002), 5830-5847, Section 1.2 and equations (3)-(4). (arXiv) (doi)
[2]
Peter J. Forrester, Meet Andréief, Bordeaux 1886, and Andreev, Kharkov 1882-83, equations (1.7) and (3.1). (arXiv)
Links
Similar tables
Selberg integrals for classical Jacobi ensembles —   the analogous normalization integrals for the Jacobi weight on the unit interval
Values of the Gamma function —   the scalar case is a Gamma integral, and the product formula uses integer and half-integer Gamma arguments
Laguerre polynomials —   the case $\beta=2,r=0$ has the one-variable weight $e^{-x}$ used in their orthogonality
Data properties
Entries are of type: real number
How well the digits are known: proven
Table is complete: no (it holds $\beta=1,2,4$ on the union of $1\leq n\leq6$, $0\leq r\leq6$; $1\leq n\leq16$, $r\in\{0,1,2\}$; and $1\leq n\leq10$, $r\in\{n,2n\}$, with intersections listed once. These noncanonical finite bounds cover small rectangular shapes, longer near-square sequences and column-to-row ratios $2$ and $3$ before extending dimension further. The 249 entries have a longest exact value of 727 characters. The attached argument-selection.md explains the coverage choices)
How they were obtained:

The generator reduces every Gamma factor to a rational times a power of $\sqrt{\pi}$, using exact rational arithmetic. The 224 rational values are stored exactly. The remaining 25 values have 100 significant digits certified by real ball arithmetic at 429 bits, with at least 427 retained relative bits. Independent 640-bit enclosures are contained in every serialized decimal interval.

more

Every value is independently obtained from the defining eigenvalue integral: a Hankel moment determinant for $\beta=2$, a confluent-Vandermonde Pfaffian for $\beta=4$, and a skew-moment Pfaffian for $\beta=1$. The moments and matrix operations use exact rational arithmetic; an odd-dimensional bordering moment carries the possible factor $\sqrt{\pi}$. The determinant and skew-kernel identities are described in [2]. Known integral controls and the excess-column recurrence are checked. The attached checks.py reproduces these checks; numerical-evidence.json records the build measurements.



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