Secondary polynomials of the Hermite polynomials in probabilist's convention $q_n$

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Polynomials

$n$

$q_n(x)$

x^2 - 2

x^3 - 5*x

x^4 - 9*x^2 + 8

x^5 - 14*x^3 + 33*x

x^6 - 20*x^4 + 87*x^2 - 48

x^7 - 27*x^5 + 185*x^3 - 279*x

x^8 - 35*x^6 + 345*x^4 - 975*x^2 + 384

10:

x^9 - 44*x^7 + 588*x^5 - 2640*x^3 + 2895*x

11:

x^10 - 54*x^8 + 938*x^6 - 6090*x^4 + 12645*x^2 - 3840

12:

x^11 - 65*x^9 + 1422*x^7 - 12558*x^5 + 41685*x^3 - 35685*x

13:

x^12 - 77*x^10 + 2070*x^8 - 23814*x^6 + 114765*x^4 - 187425*x^2 + 46080

14:

x^13 - 90*x^11 + 2915*x^9 - 42300*x^7 + 278019*x^5 - 729330*x^3 + 509985*x

15:

x^14 - 104*x^12 + 3993*x^10 - 71280*x^8 + 611415*x^6 - 2336040*x^4 + 3133935*x^2 - 645120

16:

x^15 - 119*x^13 + 5343*x^11 - 115005*x^9 + 1245915*x^7 - 6506325*x^5 + 14073885*x^3 - 8294895*x

17:

x^16 - 135*x^14 + 7007*x^12 - 178893*x^10 + 2386395*x^8 - 16288965*x^6 + 51450525*x^4 - 58437855*x^2 + 10321920

18:

x^17 - 152*x^15 + 9030*x^13 - 269724*x^11 + 4341480*x^9 - 37469520*x^7 + 162058050*x^5 - 297693900*x^3 + 151335135*x

19:

x^18 - 170*x^16 + 11460*x^14 - 395850*x^12 + 7561554*x^10 - 80424630*x^8 + 455259420*x^6 - 1223803350*x^4 + 1203216525*x^2 - 185794560

20:

x^19 - 189*x^17 + 14348*x^15 - 567420*x^13 + 12686310*x^11 - 162912750*x^9 + 1167180300*x^7 - 4302906300*x^5 + 6859400625*x^3 - 3061162125*x

21:

x^20 - 209*x^18 + 17748*x^16 - 796620*x^14 + 20603310*x^12 - 314143830*x^10 + 2775672900*x^8 - 13408094700*x^6 + 31335467625*x^4 - 27125492625*x^2 + 3715891200

22:

x^21 - 230*x^19 + 21717*x^17 - 1097928*x^15 + 32519130*x^13 - 580556340*x^11 + 6196840650*x^9 - 37918881000*x^7 + 121696499925*x^5 - 171172905750*x^3 + 68000295825*x

23:

x^22 - 252*x^20 + 26315*x^18 - 1488384*x^16 + 50044770*x^14 - 1033829160*x^12 + 13108004910*x^10 - 98983684800*x^8 + 416674583325*x^6 - 860553193500*x^4 + 664761133575*x^2 - 81749606400

24:

x^23 - 275*x^21 + 31605*x^19 - 1987875*x^17 + 75297114*x^15 - 1781769150*x^13 + 26460800730*x^11 - 241511019750*x^9 + 1288808846325*x^7 - 3659572691775*x^5 + 4601737965825*x^3 - 1645756410375*x

25:

x^24 - 299*x^22 + 37653*x^20 - 2619435*x^18 + 111018330*x^16 - 2982843630*x^14 + 51272700570*x^12 - 556103137590*x^10 + 3664417281525*x^8 - 13659762691575*x^6 + 25255014609825*x^4 - 17600023616175*x^2 + 1961990553600

26:

x^25 - 324*x^23 + 44528*x^21 - 3409560*x^19 + 160715205*x^17 - 4865271480*x^15 + 95816929320*x^13 - 1217623155840*x^11 + 9702192775275*x^9 - 45879983849700*x^7 + 116744331904200*x^5 - 132643472761800*x^3 + 43105900812975*x

27:

x^26 - 350*x^24 + 52302*x^22 - 4388538*x^20 + 228820515*x^18 - 7751748060*x^16 + 173370863700*x^14 - 2550713370660*x^12 + 24160874352615*x^10 - 141154833169350*x^8 + 471898161885150*x^6 - 789273852617250*x^4 + 500706514833525*x^2 - 51011754393600

28:

x^27 - 377*x^25 + 61050*x^23 - 5590794*x^21 + 320878635*x^19 - 12091058595*x^17 + 304733193660*x^15 - 5137770462300*x^13 + 57036699560295*x^11 - 403114038101775*x^9 + 1710657725827050*x^7 - 3941370814030650*x^5 + 4082080279402125*x^3 - 1214871076343925*x

29:

x^28 - 405*x^26 + 70850*x^24 - 7055250*x^22 + 443757699*x^20 - 18498033015*x^18 + 521782139340*x^16 - 9992154645900*x^14 + 128456673938775*x^12 - 1079618519974995*x^10 + 5662993054568850*x^8 - 17154519346814850*x^6 + 26181748152685125*x^4 - 15234653491682625*x^2 + 1428329123020800

30:

x^29 - 434*x^27 + 81783*x^25 - 8825700*x^23 + 605890725*x^21 - 27803513430*x^19 + 872422838595*x^17 - 18829417262040*x^15 + 277452017345475*x^13 - 2733682807223550*x^11 + 17353300159520325*x^9 - 66763593395799300*x^7 + 140481501759573975*x^5 - 133614981594344250*x^3 + 36659590336994625*x

31:

x^30 - 464*x^28 + 93933*x^26 - 10951200*x^24 + 817548225*x^22 - 41116244400*x^20 + 1427363829045*x^18 - 34482881442240*x^16 + 577216656722475*x^14 - 6587383025386800*x^12 + 49741855758770175*x^10 - 236653385032864800*x^8 + 655117082164019475*x^6 - 919067426174898000*x^4 + 493699195087473375*x^2 - 42849873690624000

32:

x^31 - 495*x^29 + 107387*x^27 - 13486473*x^25 + 1091144925*x^23 - 59898856875*x^21 + 2289272745375*x^19 - 61527989438685*x^17 + 1160928591845715*x^15 - 15188395563096525*x^13 + 134486022782700225*x^11 - 774605689977994875*x^9 + 2724788477433797775*x^7 - 5273993980721691225*x^5 + 4635763624512145125*x^3 - 1179297174137457375*x

33:

x^32 - 527*x^30 + 122235*x^28 - 16492329*x^26 + 1441583325*x^24 - 86060400075*x^22 + 3604992566175*x^20 - 107203631968125*x^18 + 2264380797997395*x^16 - 33659328578215725*x^14 + 345282279595077825*x^12 - 2366345074258640475*x^10 + 10297696798485471375*x^8 - 26237740609970314425*x^6 + 34045921262108881125*x^4 - 16977671416936605375*x^2 + 1371195958099968000

34:

x^33 - 560*x^31 + 138570*x^29 - 20036100*x^27 + 1886636934*x^25 - 122068182600*x^23 + 5581654843050*x^21 - 182749632565500*x^19 + 4294804449474000*x^17 - 71969972109124320*x^15 + 846499333177263150*x^13 - 6804383826087747900*x^11 + 35859684567759302250*x^9 - 116155760365285641000*x^7 + 208087722625924691550*x^5 - 169957871025837394500*x^3 + 40288002704636061375*x

35:

x^34 - 594*x^32 + 156488*x^30 - 24192090*x^28 + 2447376120*x^26 - 171082015650*x^24 + 8507708445600*x^22 - 305319379815450*x^20 + 7939727936390250*x^18 - 148958919241035750*x^16 + 1990916504836597800*x^14 - 18543981332320393950*x^12 + 116315417092553078400*x^10 - 466277451513791667750*x^8 + 1100170903364915382000*x^6 - 1327519193937539352750*x^4 + 617528830880480644125*x^2 - 46620662575398912000

36:

x^35 - 629*x^33 + 176088*x^31 - 29042040*x^29 + 3148639620*x^27 - 237114308340*x^25 + 12780094836600*x^23 - 500677299322200*x^21 + 14335965076182750*x^19 - 299277074972625750*x^17 + 4509865528655949000*x^15 - 48171457993524604200*x^13 + 354468851005624254900*x^11 - 1721366411385367246500*x^9 + 5165622516149912817000*x^7 - 8610589485844903557000*x^5 + 6566054316784789451625*x^3 - 1456700757237661060125*x

37:

x^36 - 665*x^34 + 197472*x^32 - 34675608*x^30 + 4019554860*x^28 - 325219848660*x^26 + 18939047400000*x^24 - 806954803363800*x^22 + 25327462749538950*x^20 - 585107280682674750*x^18 + 9872386621333236000*x^16 - 119844452167642125000*x^14 + 1022052178969158437100*x^12 - 5908721426717278068900*x^10 + 21951610770646412856000*x^8 - 48216742006981857309000*x^6 + 54356745298536206150625*x^4 - 23687738668934964248625*x^2 + 1678343852714360832000

38:

x^37 - 702*x^35 + 220745*x^33 - 41190864*x^31 + 5094110340*x^29 - 441719514600*x^27 + 27712276808580*x^25 - 1279818312318000*x^23 + 43852522824460350*x^21 - 1115537988501436500*x^19 + 20945638395320388750*x^17 - 286709476727912238000*x^15 + 2804396124729568792500*x^13 - 19024068913925375500200*x^11 + 85642167991905000976500*x^9 - 239344775104528631538000*x^7 + 372948556274797637759625*x^5 - 266631748389972173958750*x^3 + 55576271870507820056625*x

39:

x^38 - 740*x^36 + 246015*x^34 - 48694800*x^32 + 6411783444*x^30 - 594462599280*x^28 + 40070631057660*x^26 - 1999502113518000*x^24 + 74516805352284750*x^22 - 2077981572983916600*x^20 + 43179715061262029250*x^18 - 661860168338575206000*x^16 + 7358485307099969542500*x^14 - 57862051714753396110000*x^12 + 310173582207161567594700*x^10 - 1073505984389092320066000*x^8 + 2205184752540108215501625*x^6 - 2332188069734348007682500*x^4 + 955710341290036461504375*x^2 - 63777066403145711616000

40:

x^39 - 779*x^37 + 273393*x^35 - 57303855*x^33 + 8018227140*x^31 - 793132902540*x^29 + 57297692127060*x^27 - 3080280909052620*x^25 + 124429719532686750*x^23 - 3788229963137870250*x^21 + 86685696612818052750*x^19 - 1478740065756070367250*x^17 + 18540154899488546824500*x^15 - 167233500579206579017500*x^13 + 1052112269850251212102500*x^11 - 4413550536073387358149500*x^9 + 11539630981616724845483625*x^7 - 16877181764451455880307875*x^5 + 11354348528498951245895625*x^3 - 2231251669352950693824375*x

41:

x^40 - 819*x^38 + 302993*x^36 - 67144455*x^34 + 9966019140*x^32 - 1049604240300*x^30 + 81076196098260*x^28 - 4683106151359020*x^26 + 204409804073406750*x^24 - 6768902177229260250*x^22 + 169804959532174716750*x^20 - 3205928668206551537250*x^18 + 45014561633031555064500*x^16 - 461572912863205360717500*x^14 + 3366594338440387056502500*x^12 - 16820493824359850061937500*x^10 + 54479870357180417648123625*x^8 - 105084571866055784500372875*x^6 + 104641871317872871553195625*x^4 - 40459665320954409153999375*x^2 + 2551082656125828464640000

42:

x^41 - 860*x^39 + 334932*x^37 - 78353568*x^35 + 12315477195*x^33 - 1378351553040*x^31 + 113594645102400*x^29 - 7032311528568480*x^27 + 330701321344564170*x^25 - 11870520678069417000*x^23 + 325122388020827397000*x^21 - 6760042229332091700000*x^19 + 105642904329030440121750*x^17 - 1221719263742235780522000*x^15 + 10223167862187856796220000*x^13 - 59957096888220149758140000*x^11 + 235435442336189299332253125*x^9 - 578209442112341503165201500*x^7 + 796606323660382562645818500*x^5 - 505987954989411410235720000*x^3 + 94032401099596806911439375*x

43:

x^42 - 902*x^40 + 369330*x^38 - 91079274*x^36 + 15135544305*x^34 - 1796924356920*x^32 + 157678023195000*x^30 - 10437511764695400*x^28 + 527391779701643010*x^26 - 20455732449152500500*x^24 + 609416279464456327500*x^22 - 13891850529683429803500*x^20 + 240291908393705604686250*x^18 - 3112330852329561093231000*x^16 + 29609230202442481946355000*x^14 - 201354059102716406131245000*x^12 + 941896182959303001933628125*x^10 - 2866363997113919044386393750*x^8 + 5210158342034725511661479250*x^6 - 4900946550340072015469936250*x^4 + 1793338344579681991379413125*x^2 - 107145471557284795514880000

44:

x^43 - 945*x^41 + 406310*x^39 - 105481350*x^37 + 18504747729*x^35 - 2326489876305*x^33 + 216947139975720*x^31 - 15322081504098600*x^29 + 829781175430087650*x^27 - 34675889266968759810*x^25 + 1119848668621441258500*x^23 - 27872113214579007874500*x^21 + 530973724254985547786250*x^19 - 7654975738477870018466250*x^17 + 82143158543358620508801000*x^15 - 640950277176794248368705000*x^13 + 3520051349152769441533648125*x^11 - 12990088017570058915673278125*x^9 + 30073164352865410147765143750*x^7 - 39155018467736522209240131750*x^5 + 23550820409124372631515373125*x^3 - 4150538718839947492706773125*x

45:

x^44 - 989*x^42 + 445998*x^40 - 121731870*x^38 + 22512235785*x^36 - 2992453825725*x^34 + 296011811680200*x^32 - 22259914524678600*x^30 + 1289031693076685250*x^28 - 57881127573841052250*x^26 + 2019900896384151280500*x^24 - 54686429511015086284500*x^22 + 1142215147561056459140250*x^20 - 18227819707800916624661250*x^18 + 219085716045859308610965000*x^16 - 1943756406084263454008325000*x^14 + 12379629949672291311308428125*x^12 - 54433520067779391000752915625*x^10 + 156193180225877848100766468750*x^8 - 268401985517264444722345218750*x^6 + 239192468624087541312192568125*x^4 - 83057425880345955113400950625*x^2 + 4714400748520531002654720000

46:

x^45 - 1034*x^43 + 488523*x^41 - 140015820*x^39 + 27258896535*x^37 - 3825167473530*x^35 + 400703856113925*x^33 - 32022535823586000*x^31 + 1978525360761122250*x^29 - 95221280468194996500*x^27 + 3580315913397745471950*x^25 - 105079619598979942917000*x^23 + 2396460242217111813492750*x^21 - 42121637299275266275042500*x^19 + 563559624277363459441946250*x^17 - 5640198540535401376904370000*x^15 + 41222392422628032487900153125*x^13 - 212835830779654015869767081250*x^11 + 740747141016530499306063984375*x^9 - 1621694381396207901371776687500*x^7 + 2001168299672231040727998496875*x^5 - 1142844344290942723531592741250*x^3 + 191488643096318168174459510625*x

47:

x^46 - 1080*x^44 + 534017*x^42 - 160531728*x^40 + 32858562555*x^38 - 4860730319640*x^36 + 538356732097275*x^34 - 45639079160875200*x^32 + 3002481428896337850*x^30 - 154516738349722518000*x^28 + 6242847781794433875450*x^26 - 197995060832650901820000*x^24 + 4912035999723805782579750*x^22 - 94663534087083863395494000*x^20 + 1402039330836205624176363750*x^18 - 15718141478644929573008760000*x^16 + 130635187102504151372283103125*x^14 - 782298808464579416189954775000*x^12 + 3244689064134382485340698103125*x^10 - 8806580671786588914007034250000*x^8 + 14347659633466395497955878559375*x^6 - 12145697900998969623892450875000*x^4 + 4012130233592232103390903239375*x^2 - 216862434431944426122117120000

48:

x^47 - 1127*x^45 + 582615*x^43 - 183492309*x^41 + 39439306095*x^39 - 6141898456785*x^37 + 718139603353185*x^35 - 64472160398229675*x^33 + 4507540612604879850*x^31 - 247507430305495263750*x^29 + 10718247963799598710950*x^27 - 366269908762344939001650*x^25 + 9850778120875863099678750*x^23 - 207297165471288118629653250*x^21 + 3381756283902143139103361250*x^19 - 42205443819681012166780233750*x^17 + 395724518507668016086788493125*x^15 - 2719751252328096943121261971875*x^13 + 13247973110778121231219750921875*x^11 - 43621696299563522381392041515625*x^9 + 90567295559088166862429382871875*x^7 - 106200607985593828538108380228125*x^5 + 57725814415266540109375762078125*x^3 - 9216828659958898330321714119375*x

49:

x^48 - 1175*x^46 + 634455*x^44 - 209125125*x^42 + 47144829039*x^40 - 7719109459425*x^38 + 951454658695905*x^36 - 90313283538898875*x^34 + 6698216412326889450*x^32 - 391626538892519480550*x^30 + 18135051404586279574950*x^28 - 665926602288477765023250*x^26 + 19354541040843106387038750*x^24 - 443074893458030796193481250*x^22 + 7925605920082168582087073250*x^20 - 109503331699818882127245693750*x^18 + 1150195309482624635591208973125*x^16 - 8990240233248296208990850921875*x^14 + 50798315917077933208337580121875*x^12 - 199366771378013881677745550465625*x^10 + 513283167804844434734767026871875*x^8 - 794888270391980812439990551078125*x^6 + 640719313663217082056213404078125*x^4 - 201799079872386039293085069609375*x^2 + 10409396852733332453861621760000

50:

x^49 - 1224*x^47 + 689678*x^45 - 237673260*x^43 + 56135952180*x^41 - 9651635458080*x^39 + 1252407683078370*x^37 - 125502124103204940*x^35 + 9857352271840143525*x^33 - 612496028910158593200*x^31 + 30262915489555547498700*x^29 - 1191120752514658101859800*x^27 + 37301766570198008398119600*x^25 - 925763021380948088077740000*x^23 + 18083167028175286394940082500*x^21 - 275209389611023895943310395000*x^19 + 3218262056646994231763440426875*x^17 - 28380741640124028997243487085000*x^15 + 184066127281154683421279416743750*x^13 - 848517453806141822007513345637500*x^11 + 2650746286483457031422977061137500*x^9 - 5232685752787300988699030311800000*x^7 + 5844549104957314680423524035256250*x^5 - 3030363986220446504652497411437500*x^3 + 462034001190719350639625613609375*x

Definition

For $n\geq 0$, let $He_n$ be the Hermite polynomial in probabilist's convention. The secondary polynomial $q_n$ is $q_n(x)=\int_{-\infty}^{\infty}\frac{He_n(t)-He_n(x)}{t-x}\,\frac{e^{-t^2/2}}{\sqrt{2\pi}}\,\mathrm{d}t$ [1], using the standard normal probability density.

Parameters

$n$

— integer ($n\geq 0$)

Formulas

(1)

$q_0(x)=0$, $q_1(x)=1$, and $q_{n+1}(x)=xq_n(x)-nq_{n-1}(x)$ for $n\geq 1$.

(2)

If $He_n(x)=\sum_{j=0}^{n}c_jx^j$, then $q_n(x)=\sum_{j=1}^{n}c_j\sum_{i=0}^{j-1}m_i x^{j-1-i}$, where $m_i=0$ for odd $i$ and $m_{2r}=(2r-1)!!$ for $r\geq0$.

(3)

$q_n(z)/He_n(z)$ is the $n$-th approximant of the continued fraction $\dfrac{1}{z-}\dfrac{1}{z-}\dfrac{2}{z-}\cdots\dfrac{n-1}{z}$ [3].

(4)

At a root $y_k$ of $He_n$, $q_n(y_k)/He_n'(y_k)$ is the weight of the $n$-point Gauss-Hermite rule for the standard normal probability density.

Comments

(5)

$He_n$ is the Hermite polynomial in probabilist's convention, with leading coefficient $1$ and orthogonal for the weight $e^{-x^2/2}$ on the real line [2]. The quotient $(He_n(t)-He_n(x))/(t-x)$ is a polynomial in $t$ and $x$, so integrating against the standard normal moments gives $q_0=0$ and, for $n\geq 1$, an integral polynomial of degree $n-1$.

(6)

$q_n$ relates to the secondary polynomials of the Hermite polynomials in physicist's convention $q^H_n$ via $q_n(x)=2^{-(n+1)/2}q^H_n(x/\sqrt{2})$.

(7)

Wikipedia [1] defines the secondary polynomials for an arbitrary density and does not choose one. This table uses the probability density $e^{-t^2/2}/\sqrt{2\pi}$, so $m_0=1$, $q_0=0$ and $q_1=1$. Using the unnormalised Hermite weight $e^{-t^2/2}$ would multiply every entry by $\sqrt{2\pi}$, so the values would not be polynomials over $\mathbb{Z}$ or $\mathbb{Q}$.

(8)

These are also the numerator polynomials for the monic recurrence of $He_n$ [3]; in the notation of [4] they are the order-one associated monic Hermite polynomials with the index shifted by one.

Programs

(P1)

Sage

R.<x> = ZZ[]
q = [R(0), R(1)]
for n in range(1, 51):
    q.append(x*q[n] - n*q[n-1])

q[51]        # the next one after this table

Links

[1]

Wikipedia: Secondary polynomials

[2]

DLMF 18.3: classical orthogonal polynomial definitions

[3]

DLMF 18.2(x): orthogonal polynomials and continued fractions

[4]

DLMF 18.30: associated orthogonal polynomials

Similar tables

Hermite polynomials in probabilist's convention — the $He_n$ these are the secondary polynomials of, and the recurrence they share

Nodes and weights of Gauss-Hermite quadrature — the same rule in probabilists' form: if $x_k,w_k$ are its nodes and weights, then $y_k=\sqrt{2}\,x_k$ and $q_n(y_k)/He_n'(y_k)=w_k/\sqrt{\pi}$

Secondary polynomials of the Legendre polynomials — the same construction for the Legendre weight

Data properties

Entries are of type: integral polynomial

How well the digits are known: exact

Table is complete: no (it holds every $n\leq 50$, matching the range of the Hermite polynomial table in probabilist's convention)

How they were obtained:

Every value is exact, a polynomial with integer coefficients, so there is no precision to choose. Each entry was built from the recurrence and checked, before it was written, against the defining integral evaluated exactly from the standard normal moments.

For $n\geq 1$, it was also checked that $q_n$ has degree $n-1$, is an even or odd function according to the parity of $n-1$, and is monic.