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

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Polynomials

$n$

$q_n(x)$

4*x

8*x^2 - 8

16*x^3 - 40*x

32*x^4 - 144*x^2 + 64

64*x^5 - 448*x^3 + 528*x

128*x^6 - 1280*x^4 + 2784*x^2 - 768

256*x^7 - 3456*x^5 + 11840*x^3 - 8928*x

512*x^8 - 8960*x^6 + 44160*x^4 - 62400*x^2 + 12288

10:

1024*x^9 - 22528*x^7 + 150528*x^5 - 337920*x^3 + 185280*x

11:

2048*x^10 - 55296*x^8 + 480256*x^6 - 1559040*x^4 + 1618560*x^2 - 245760

12:

4096*x^11 - 133120*x^9 + 1456128*x^7 - 6429696*x^5 + 10671360*x^3 - 4567680*x

13:

8192*x^12 - 315392*x^10 + 4239360*x^8 - 24385536*x^6 + 58759680*x^4 - 47980800*x^2 + 5898240

14:

16384*x^13 - 737280*x^11 + 11939840*x^9 - 86630400*x^7 + 284691456*x^5 - 373416960*x^3 + 130556160*x

15:

32768*x^14 - 1703936*x^12 + 32710656*x^10 - 291962880*x^8 + 1252177920*x^6 - 2392104960*x^4 + 1604574720*x^2 - 165150720

16:

65536*x^15 - 3899392*x^13 + 87539712*x^11 - 942120960*x^9 + 5103267840*x^7 - 13324953600*x^5 + 14411658240*x^3 - 4246986240*x

17:

131072*x^16 - 8847360*x^14 + 229605376*x^12 - 2930982912*x^10 + 19549347840*x^8 - 66719600640*x^6 + 105370675200*x^4 - 59840363520*x^2 + 5284823040

18:

262144*x^17 - 19922944*x^15 + 591790080*x^13 - 8838316032*x^11 + 71130808320*x^9 - 306950307840*x^7 + 663789772800*x^5 - 609677107200*x^3 + 154967178240*x

19:

524288*x^18 - 44564480*x^16 + 1502085120*x^14 - 25942425600*x^12 + 247777001472*x^10 - 1317677137920*x^8 + 3729485168640*x^6 - 5012698521600*x^4 + 2464187443200*x^2 - 190253629440

20:

1048576*x^19 - 99090432*x^17 + 3761242112*x^15 - 74372874240*x^13 + 831410012160*x^11 - 5338324992000*x^9 + 19123082035200*x^7 - 35249408409600*x^5 + 28096104960000*x^3 - 6269260032000*x

21:

2097152*x^20 - 219152384*x^18 + 9305063424*x^16 - 208829153280*x^14 + 2700517048320*x^12 - 20587730042880*x^10 + 90953249587200*x^8 - 219678223564800*x^6 + 256700150784000*x^4 - 111106017792000*x^2 + 7610145177600

22:

4194304*x^21 - 482344960*x^19 + 22771924992*x^17 - 575630475264*x^15 + 8524694814720*x^13 - 76094680596480*x^11 + 406116148838400*x^9 - 1242525892608000*x^7 + 1993875454771200*x^5 - 1402248443904000*x^3 + 278529211699200*x

23:

8388608*x^22 - 1056964608*x^20 + 55186554880*x^18 - 1560683741184*x^16 + 26237872373760*x^14 - 271012111319040*x^12 + 1718092419563520*x^10 - 6486994767052800*x^8 + 13653592746393600*x^6 - 14099303522304000*x^4 + 5445723206246400*x^2 - 334846387814400

24:

16777216*x^23 - 2306867200*x^21 + 132560977920*x^19 - 4168876032000*x^17 + 78954746609664*x^15 - 934160184115200*x^13 + 6936540146565120*x^11 - 31655332380672000*x^9 + 84463376552755200*x^7 - 119916877964083200*x^5 + 75394874832076800*x^3 - 13482036513792000*x

25:

33554432*x^24 - 5016387584*x^22 + 315856257024*x^20 - 10986706698240*x^18 + 232822312796160*x^16 - 3127738242170880*x^14 + 26881661636444160*x^12 - 145779100900392960*x^10 + 480302501924044800*x^8 - 895206207755059200*x^6 + 827556318734745600*x^4 - 288358786927411200*x^2 + 16072626615091200

26:

67108864*x^25 - 10871635968*x^23 + 747055874048*x^21 - 28601462292480*x^19 + 674088427192320*x^17 - 10203213814824960*x^15 + 100471332478648320*x^13 - 638385209129041920*x^11 + 2543371622881689600*x^9 - 6013581243147878400*x^7 + 7650956535673651200*x^5 - 4346461315458662400*x^3 + 706247078919782400*x

27:

134217728*x^26 - 23488102400*x^24 + 1754963902464*x^22 - 73627449950208*x^20 + 1919485602693120*x^18 - 32513187895050240*x^16 + 363585053550182400*x^14 - 2674616823353180160*x^12 + 12667256492583813120*x^10 - 37002892586346086400*x^8 + 61852635874610380800*x^6 - 51725851205124096000*x^4 + 16407151078064947200*x^2 - 835776583984742400

28:

268435456*x^27 - 50600083456*x^25 + 4096996147200*x^23 - 187595917099008*x^21 + 5383450169180160*x^19 - 101427150858485760*x^17 + 1278143653100912640*x^15 - 10774685600553369600*x^13 + 59807314278135889920*x^11 - 211347852808303411200*x^9 + 448438658879206195200*x^7 - 516603355336625356800*x^5 + 267523213190897664000*x^3 - 39808895429637734400*x

29:

536870912*x^28 - 108716359680*x^26 + 9509326028800*x^24 - 473469812736000*x^22 + 14890037535571968*x^20 - 310345495467786240*x^18 + 4377025828324638720*x^16 - 41910134199916953600*x^14 + 269393170664049868800*x^12 - 1132062069201300357120*x^10 + 2969039302593793228800*x^8 - 4496954319651432038400*x^6 + 3431694093868744704000*x^4 - 998418251230912512000*x^2 + 46803488703145574400

30:

1073741824*x^29 - 233001975808*x^27 + 21953456898048*x^25 - 1184565402009600*x^23 + 40660638262886400*x^21 - 932931100748021760*x^19 + 14636826406441451520*x^17 - 157952600279686840320*x^15 + 1163718106160195174400*x^13 - 5732948366534482329600*x^11 + 18196254068069184307200*x^9 - 35003350854296823398400*x^7 + 36826382797261760102400*x^5 - 17513182867533889536000*x^3 + 2402522912325279744000*x

31:

2147483648*x^30 - 498216206336*x^28 + 50429895376896*x^26 - 2939690365747200*x^24 + 109729465289932800*x^22 - 2759264453630361600*x^20 + 47894382540950077440*x^18 - 578526750258852003840*x^16 + 4842044264315407564800*x^14 - 27629486972911956787200*x^12 + 104316232288216390041600*x^10 - 248149059864221240524800*x^8 + 343470024773609442508800*x^6 - 240928011367192461312000*x^4 + 64710140898505310208000*x^2 - 2808209322188734464000

32:

4294967296*x^31 - 1063004405760*x^29 + 115305913253888*x^27 - 7240495059173376*x^25 + 292901985504460800*x^23 - 8039488479559680000*x^21 + 153630493328277504000*x^19 - 2064536737717074001920*x^17 + 19477149745971399229440*x^15 - 127409496527756014387200*x^13 + 564075263301570684518400*x^11 - 1624465871948731908096000*x^9 + 2857147802513621935718400*x^7 - 2765091756164614048972800*x^5 + 1215237619584111771648000*x^3 - 154572839208544813056000*x

33:

8589934592*x^32 - 2263447764992*x^30 + 262497663713280*x^28 - 17708503422468096*x^26 + 773944154416742400*x^24 - 23101662737675059200*x^22 + 483853911688898150400*x^20 - 7194313958054952960000*x^18 + 75980011508509326704640*x^16 - 564709825971698112921600*x^14 + 2896437692869506603417600*x^12 - 9925170610343312778854400*x^10 + 21595835436337403265024000*x^8 - 27512265097840232418508800*x^6 + 17849867966668541067264000*x^4 - 4450594695921429479424000*x^2 + 179725396620079005696000

34:

17179869184*x^33 - 4810363371520*x^31 + 595153618206720*x^29 - 43027197119692800*x^27 + 2025760982738927616*x^25 - 65534856518644531200*x^23 + 1498314063028735180800*x^21 - 24528240475776221184000*x^19 + 288219447706345537536000*x^17 - 2414911535177508574986240*x^15 + 14201902156570910156390400*x^13 - 57079308598590290735923200*x^11 + 150406418421291112464384000*x^9 - 243596285161579512594432000*x^7 + 218195791840201609366732800*x^5 - 89106872284394235887616000*x^3 + 10561258181004115673088000*x

35:

34359738368*x^34 - 10204842295296*x^32 + 1344221684432896*x^30 - 103904235371888640*x^28 + 5255700198205685760*x^26 - 183697915537627545600*x^24 + 4567541192219374387200*x^22 - 81958546946397516595200*x^20 + 1065652244560427876352000*x^18 - 9996463852933651365888000*x^16 + 66804072479217291991449600*x^14 - 311116380312307030504243200*x^12 + 975724438345927493890867200*x^10 - 1955709379994102447210496000*x^8 + 2307225610333539023192064000*x^6 - 1392004766302249264349184000*x^4 + 323762955684665435947008000*x^2 - 12221326970165372387328000

36:

68719476736*x^35 - 21612275433472*x^33 + 3025168804872192*x^31 - 249469224018247680*x^29 + 13523304194789867520*x^27 - 509199099866980024320*x^25 + 13722522340743865958400*x^23 - 268799078304806495846400*x^21 + 3848281322425191235584000*x^19 - 40168289045311490359296000*x^17 + 302651952420860184231936000*x^15 - 1616365911584577771955814400*x^13 + 5947000478593175339296358400*x^11 - 14439868049478582766927872000*x^9 + 21666191181977643927994368000*x^7 - 18057714961418611184369664000*x^5 + 6885006971276927384027136000*x^3 - 763730726610618841890816000*x

37:

137438953472*x^36 - 45698452029440*x^34 + 6785086255005696*x^32 - 595722409315663872*x^30 + 34527713336355717120*x^28 - 1396808614004769423360*x^26 + 40671294600196915200000*x^24 - 866461122449407947571200*x^22 + 13597578024991003666022400*x^20 - 157063539698973787815936000*x^18 + 1325049302252943266807808000*x^16 - 8042625041672800567296000000*x^14 + 34294380339672456874898227200*x^12 - 99131895659863945093998182400*x^10 + 184143457723530664055144448000*x^8 - 202235673866852032038567936000*x^6 + 113994357116315801801195520000*x^4 - 24838394262517149071966208000*x^2 + 879935541851906811887616000

38:

274877906944*x^37 - 96482145337344*x^35 + 15169480892088320*x^33 - 1415307310191869952*x^31 + 87516149250061762560*x^29 - 3794341738423989043200*x^27 + 119023322590550352199680*x^25 - 2748388898113861976064000*x^23 + 47086287844537688024678400*x^21 - 598899897257411727065088000*x^19 + 5622551993858936820203520000*x^17 - 38481494562489254767755264000*x^15 + 188199838136603668874526720000*x^13 - 638341826735622865295926886400*x^11 + 1436837151108476452862951424000*x^9 - 2007769495200049714748719104000*x^7 + 1564259621377608831245746176000*x^5 - 559167304399526924561940480000*x^3 + 58275944852889607923695616000*x

39:

549755813888*x^38 - 203409651138560*x^36 + 33812044138414080*x^34 - 3346281175764172800*x^32 + 220307201608113979392*x^30 - 10212789690411012587520*x^28 + 344204099845463180574720*x^26 - 8587796185842689507328000*x^24 + 160023620995230380064768000*x^22 - 2231215724414139732747878400*x^20 + 23181933004839881514418176000*x^18 - 177666736096202197812903936000*x^16 + 987639179440340180863549440000*x^14 - 3883056559286352453084119040000*x^12 + 10407698372366612732869764710400*x^10 - 18010441777388429897688416256000*x^8 + 18498430456635972097422655488000*x^6 - 9781905749639054786014740480000*x^4 + 2004269853657082545316823040000*x^2 - 66875101180744917703458816000

40:

1099511627776*x^39 - 428259779018752*x^37 + 75149695613140992*x^35 - 7875781861111234560*x^33 + 551008373411193815040*x^31 - 27251839022326842654720*x^29 + 984366855287997506519040*x^27 - 26459411533748306486231040*x^25 + 534421576043341994262528000*x^23 - 8135161900702219131420672000*x^21 + 93078057995757877739913216000*x^19 - 793892527713401467601682432000*x^17 + 4976834934754842233612009472000*x^15 - 22445700493227791078381322240000*x^13 + 70606059230111808958821826560000*x^11 - 148094181341238023118687043584000*x^9 + 193602881538875821945245401088000*x^7 - 141576061966731598409197682688000*x^5 + 47623589450477265206465003520000*x^3 - 4679273900886879253455175680000*x

41:

2199023255552*x^40 - 900500023148544*x^38 + 166572163317366784*x^36 - 18456527253295595520*x^34 + 1369719240883521454080*x^32 - 72128254173302803660800*x^30 + 2785756885808876020039680*x^28 - 80455151055133667418439680*x^26 + 1755866846954099149111296000*x^24 - 29072213481022868668022784000*x^22 + 364653373944646934099656704000*x^20 - 3442339695813993456356818944000*x^18 + 24167008757205860292256333824000*x^16 - 123902535341682796625846599680000*x^14 + 451856643203131814164373176320000*x^12 - 1128804232471805064866955264000000*x^10 + 1828041105268826035705564102656000*x^8 - 1763026560464340964612207804416000*x^6 + 877799638872078913294109245440000*x^4 - 169700136094340362132256194560000*x^2 + 5350008094459593416276705280000

42:

4398046511104*x^41 - 1891159999774720*x^39 + 368261628514271232*x^37 - 43075329546868752384*x^35 + 3385252594378164142080*x^33 - 189439194966323500154880*x^31 + 7806164571448553137766400*x^29 - 241628384243883130371440640*x^27 + 5681405439675559430093537280*x^25 - 101966996197599780865572864000*x^23 + 1396390023746875836975808512000*x^21 - 14517080147280132887386521600000*x^19 + 113433204786910640927850627072000*x^17 - 655905535333262656407877976064000*x^15 + 2744260726850942496756014776320000*x^13 - 8047305321812778464357300305920000*x^11 + 15799805080519169967143465779200000*x^9 - 19401489407116499328734538498048000*x^7 + 13364836359016148896142428471296000*x^5 - 4244534607127816471194642677760000*x^3 + 394400476061643285615877816320000*x

43:

8796093022208*x^42 - 3967037953015808*x^40 + 812165258973020160*x^38 - 100142720812396314624*x^36 + 8320853478033158307840*x^34 - 493934806166862802452480*x^32 + 21671102493454541783040000*x^30 - 717260346895711846426214400*x^28 + 18121043567982346923336007680*x^26 - 351426807539343890256494592000*x^24 + 5234845979899672642065530880000*x^22 - 59665043705910608239144206336000*x^20 + 516022944022196732189674045440000*x^18 - 3341839806271817577365287993344000*x^16 + 15896334422403239910083143925760000*x^14 - 54050568672688629318521947422720000*x^12 + 126419165688669965383111173734400000*x^10 - 192358431656814385656736461619200000*x^8 + 174823903797036938819710312513536000*x^6 - 82224238879510261659094461972480000*x^4 + 15043612384047876990341275975680000*x^2 - 449400679934605846967243243520000

44:

17592186044416*x^43 - 8312307905986560*x^41 + 1786970277926666240*x^39 - 231955941677019955200*x^37 + 20346185297097029320704*x^35 - 1279001335450247721123840*x^33 + 59633975754012904579399680*x^31 - 2105850846935999262700339200*x^29 + 57022128180938643058615910400*x^27 - 1191454482890785891501033390080*x^25 + 19238853632792926438570328064000*x^23 - 239419629454052538458207944704000*x^21 + 2280514780710484892694588948480000*x^19 - 16438935224217950274525729914880000*x^17 + 88200544883467068271243793793024000*x^15 - 344107559854558313358061165608960000*x^13 + 944906589053238878700950173777920000*x^11 - 1743500100238277388487810980249600000*x^9 + 2018175896606092819910590935859200000*x^7 - 1313824404634409328386437772476416000*x^5 + 395117200981087970503421822238720000*x^3 - 34817242301170534256899978690560000*x

45:

35184372088832*x^44 - 17398671997927424*x^42 + 3923039895718723584*x^40 - 535382426143665684480*x^38 + 49504930025684934328320*x^36 - 3290237776967413373337600*x^34 + 162734214450709735774617600*x^32 - 6118758713295998202308198400*x^30 + 177163166888699928602738688000*x^28 - 3977560799764018312255635456000*x^26 + 69403266329048115219712180224000*x^24 - 939505705139276269418182606848000*x^22 + 9811553407541103310433868054528000*x^20 - 78287889522389212981742775828480000*x^18 + 470482992718854083350632931000320000*x^16 - 2087092548880601738803438996684800000*x^14 + 6646263221303077137431831720755200000*x^12 - 14611886781079511730689405249126400000*x^10 + 20963893779011851577613990494208000000*x^8 - 18012152343408069272907383046144000000*x^6 + 8025967423359078967007156298055680000*x^4 - 1393472374398554243663832243240960000*x^2 + 39547259834245314533117405429760000

46:

70368744177664*x^45 - 36380640739852288*x^43 + 8594187502976237568*x^41 - 1231592177300731330560*x^39 + 119885894802301664624640*x^37 - 8411632230673559385538560*x^35 + 440578549091941766450380800*x^33 - 17604575244453157816762368000*x^31 + 543852910001639790848507904000*x^29 - 13087113135812514499786702848000*x^27 + 246037436118266960674517365555200*x^25 - 3610508237229915918307694739456000*x^23 + 41170873465946935082106451132416000*x^21 - 361822109308722066305998557020160000*x^19 + 2420470155617323691408581554339840000*x^17 - 12112234137273239622018819434741760000*x^15 + 44262206829516402477089168346316800000*x^13 - 114265366576950522544464326138265600000*x^11 + 198842796579468668119130969210880000000*x^9 - 217660135381364492337767950319616000000*x^7 + 134296131263814997488793712118988800000*x^5 - 38347492837095025832635628487966720000*x^3 + 3212646326773838712187232893009920000*x

47:

140737488355328*x^46 - 75998243711877120*x^44 + 18789052829761798144*x^42 - 2824104025007585230848*x^40 + 289026972809820572221440*x^38 - 21377718023710132729282560*x^36 + 1183858973664885563247820800*x^34 - 50180698218371611324789555200*x^32 + 1650631621626511416309370060800*x^30 - 42473237625385422431025364992000*x^28 + 858010465814823606076553193062400*x^26 - 13606116976732258436828910059520000*x^24 + 168776271804707286950685702094848000*x^22 - 1626307132111225637171912975056896000*x^20 + 12043426147294454977137498484899840000*x^18 - 67508903604681054912295868521512960000*x^16 + 280536928156048164924094724387635200000*x^14 - 839986949513784141732657170586009600000*x^12 + 1741979177018252415461687221341388800000*x^10 - 2363998498431819329816023026106368000000*x^8 + 1925710278121172368085066664483225600000*x^6 - 815083988623225316629929636397056000000*x^4 + 134624751098214667841447032164188160000*x^2 - 3638347904750568937046801299537920000

48:

281474976710656*x^47 - 158611149376454656*x^45 + 40997885889069711360*x^43 - 6456061675294936793088*x^41 + 693823610285909889515520*x^39 - 54024710158836621933281280*x^37 + 3158411377013085708736266240*x^35 - 141775780051385748695914905600*x^33 + 4956093316231619667394402713600*x^31 - 136068648790924985201810472960000*x^29 + 2946209566396023575133056453836800*x^27 - 50339752948581611177062452481228800*x^25 + 676940317909026670222294709698560000*x^23 - 7122676370021463172061832356560896000*x^21 + 58098130569608784187038861329694720000*x^19 - 362542001837390536816998403150970880000*x^17 + 1699623865215780854317958475640995840000*x^15 - 5840621341002110116311696165725798400000*x^13 + 14224902812269853950103021099679744000000*x^11 - 23419219875333693462830357158035456000000*x^9 + 24311473282090607015920320659010355200000*x^7 - 14254004316045060397206468211979059200000*x^5 + 3873913828883361763950643142197248000000*x^3 - 309265450526241976819693494542008320000*x

49:

562949953421312*x^48 - 330733097635020800*x^46 + 89291603174479626240*x^44 - 14715872422247006208000*x^42 + 1658761206972547961192448*x^40 - 135796009707416018799820800*x^38 + 8369083684302344159483658240*x^36 - 397202021574600511463620608000*x^34 + 14729533661426914021968602726400*x^32 - 430597933257995066369320471756800*x^30 + 9969849944829047703644547946905600*x^28 - 183048510615386288537474011496448000*x^26 + 2660067865584350150380164785111040000*x^24 - 30447874833294825891389492114227200000*x^22 + 272321745821895229542581368264851456000*x^20 - 1881252913815048751439196660852326400000*x^18 + 9880102476480942980216320329347235840000*x^16 - 38612787784984844065336485872664576000000*x^14 + 109088552777862985506541130575616409600000*x^12 - 214068440744419618809982687564844236800000*x^10 + 275566802413635869694178851824232038400000*x^8 - 213376195331722668130579336214347776000000*x^6 + 85995890565596353924374531378511872000000*x^4 - 13542507006481092066418302076846080000000*x^2 + 349281398856054617956492924755640320000

50:

1125899906842624*x^49 - 689050742987685888*x^47 + 194127098987851808768*x^45 - 33449537661622844129280*x^43 + 3950216458123999728107520*x^41 - 339586733222851206772162560*x^39 + 22032588964170677268428881920*x^37 - 1103928358096483422383395307520*x^35 + 43353093767889631416136866201600*x^33 - 1346893011506688860143292409446400*x^31 + 33274427471168743957066522243891200*x^29 - 654825558737582887437987555468902400*x^27 + 10253431520129698196112449913382502400*x^25 - 127235900821674265464563865778913280000*x^23 + 1242665775905893849947222307472670720000*x^21 - 9456122623451758353208201732014735360000*x^19 + 55289321133026158568498484167489617920000*x^17 - 243788714361116211853832902358146744320000*x^15 + 790557996973932762411628485392361062400000*x^13 - 1822177357091284924730061442898303385600000*x^11 + 2846217152609973698745732705135938764800000*x^9 - 2809276772308324823821350097011710361600000*x^7 + 1568884204103608626774982947530971545600000*x^5 - 406728569243531636999999632089022464000000*x^3 + 31006576949283822964242948314628096000000*x

Definition

For $n\geq0$, let $H_n$ be the Hermite polynomial in physicist's convention. The secondary polynomial $q_n$ is $q_n(x)=\int_{-\infty}^{\infty}(H_n(t)-H_n(x))/(t-x)\,\rho(t)\,dt$ [1], where $\rho(t)=e^{-t^2}/\sqrt{\pi}$ is the probability density proportional to the Hermite weight.

Parameters

$n$

— degree ($n\geq0$)

Formulas

(1)

$q_0=0$, $q_1=2$, and $q_{n+1}(x)=2xq_n(x)-2nq_{n-1}(x)$ for $n\geq1$.

(2)

If $H_n(x)=\sum_{j=0}^{n}c_jx^j$, then $q_n(x)=\sum_{j=1}^{n}c_j\sum_{i=0}^{j-1}m_i x^{j-1-i}$, where $m_i=\int_{-\infty}^{\infty}t^i\rho(t)\,dt$ is $0$ for odd $i$ and $(i-1)!!/2^{i/2}$ for even $i$.

(3)

Let $F(z)=\int_{-\infty}^{\infty}\rho(t)/(z-t)\,dt$. Then $q_n(z)/H_n(z)$ is the $n$-th convergent of the Stieltjes continued fraction for $F(z)$.

(4)

At a root $x_k$ of $H_n$, $q_n(x_k)/H_n'(x_k)=w_k/\sqrt{\pi}$, the normalised Gauss-Hermite quadrature weight at $x_k$.

Comments

(5)

The weight is normalised to a probability density, so $m_0=1$. With the unnormalised weight $e^{-t^2}$ the same integral is $\sqrt{\pi}$ times this table's polynomial, which is not a polynomial over $\mathbb{Z}$ or $\mathbb{Q}$.

(6)

Equivalently, $q_n$ is the numerator polynomial for the recurrence of $H_n$. Since $H_0=1$, $H_1=2x$ and $H_{n+1}=2xH_n-2nH_{n-1}$, the corresponding numerator sequence has $q_0=0$, $q_1=2$ and the same recurrence. This is the zeroth corecursive polynomial, or the order-one associated monic polynomial after undoing the monic rescaling, in the terminology of [3].

(7)

At the roots $x_1<\cdots<x_n$ of $H_n$, $q_n(x_k)/H_n'(x_k)$ is $w_k/\sqrt{\pi}$, where $w_k$ is the weight of the $n$-point Gauss-Hermite quadrature rule for the unnormalised weight $e^{-x^2}$.

(8)

The first values are $q_0=0$, $q_1=2$, $q_2=4x$, $q_3=8x^2-8$ and $q_4=16x^3-40x$.

Programs

(P1)

Sage

import numberdb.sage as numberdb
from sage.rings.integer_ring import ZZ
from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing

R = PolynomialRing(ZZ, "x")
x = R.gen()
q_prev = R.zero()
q = R(2)
for n in range(1, 10):
    q_prev, q = q, 2*x*q - 2*n*q_prev
q

Links

[1]

Wikipedia: Secondary polynomials

[2]

Wikipedia: Hermite polynomials

[3]

DLMF 18.30: associated orthogonal polynomials

[4]

Requested in numberdb-data#93

Similar tables

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

Nodes and weights of Gauss-Hermite quadrature — the weights divided by $\sqrt{\pi}$ are $q_n(x_k)/H_n'(x_k)$ at the nodes

Secondary polynomials of the Legendre polynomials — the same construction for the Legendre weight, in the unnormalised convention already used there

Data properties

Entries are of type: integral polynomial

How well the digits are known: exact

Table is complete: no (it holds every $n\leq50$, matching the range of the physicists' Hermite polynomial table)

How they were obtained:

Every value is exact, a polynomial with integer coefficients, so there is no precision to choose. The generator builds $q_n$ from the exact recurrence (1).

Before the values were written, the recurrence values were checked against the defining integral evaluated exactly from the moments in (2), against a direct polynomial long-division computation of that integral, against the degree and parity forced by the definition, and against the normalised Gauss-Hermite weight identity (4) through the stored range.