Secondary polynomials of the Legendre polynomials $q_n$

edit · history · discussion · files · long url · orthogonal polynomials polynomial

Polynomials

$n$

$q_n(x)$

comment: $q_0=0$: $P_0$ is constant, so the integrand vanishes.
equals: Zero

comment: $q_1=\int_{-1}^{1}dt=2$, the length of the interval; with the probability density $\frac12$ it would be $1$.
equals: Integers#2

3*x

comment: $q_2=3x$ is also the Gegenbauer polynomial $C_1^{(3/2)}$;
equals: Gegenbauer_polynomials#3/2,1

5*x^2 - 4/3

35/4*x^3 - 55/12*x

63/4*x^4 - 49/4*x^2 + 16/15

231/8*x^5 - 119/4*x^3 + 231/40*x

429/8*x^6 - 275/4*x^4 + 849/40*x^2 - 32/35

6435/64*x^7 - 9867/64*x^5 + 4213/64*x^3 - 15159/2240*x

12155/64*x^8 - 65065/192*x^6 + 11869/64*x^4 - 14179/448*x^2 + 256/315

10:

46189/128*x^9 - 70499/96*x^7 + 157157/320*x^5 - 26741/224*x^3 + 61567/8064*x

11:

88179/128*x^10 - 12597/8*x^8 + 1195831/960*x^6 - 793/2*x^4 + 116519/2688*x^2 - 512/693

12:

676039/512*x^11 - 5143775/1536*x^9 + 3917667/1280*x^7 - 1548989/1280*x^5 + 887107/4608*x^3 - 995215/118272*x

13:

1300075/512*x^12 - 10868627/1536*x^10 + 1878891/256*x^8 - 4450617/1280*x^6 + 3392503/4608*x^4 - 949477/16896*x^2 + 2048/3003

14:

5014575/1024*x^13 - 7614725/512*x^11 + 53035631/3072*x^9 - 17109633/1792*x^7 + 13023037/5120*x^5 - 14554567/50688*x^3 + 4010263/439296*x

15:

9694845/1024*x^14 - 15935205/512*x^12 + 40941219/1024*x^10 - 45354045/1792*x^8 + 25076105/3072*x^6 - 6996503/5632*x^4 + 30793817/439296*x^2 - 4096/6435

16:

300540195/16384*x^15 - 1063201335/16384*x^13 + 1497619539/16384*x^11 - 7480148665/114688*x^9 + 8521248725/344064*x^7 - 863543407/180224*x^5 + 948900679/2342912*x^3 - 1032332599/105431040*x

17:

583401555/16384*x^16 - 2209854375/16384*x^14 + 3387074691/16384*x^12 - 18835974729/114688*x^10 + 8245663415/114688*x^8 - 834786679/49152*x^6 + 4581320375/2342912*x^4 - 995524399/11714560*x^2 + 65536/109395

18:

2268783825/32768*x^17 - 1145195645/4096*x^15 + 948764145/2048*x^13 - 830829207/2048*x^11 + 69293457805/344064*x^9 - 1617518355/28672*x^7 + 443420841/53248*x^5 - 5013017/9152*x^3 + 4147223271/398295040*x

19:

4418157975/32768*x^18 - 4736583775/8192*x^16 + 8437146465/8192*x^14 - 8076126045/8192*x^12 + 26924816347/49152*x^10 - 10205648895/57344*x^8 + 3440516985/106496*x^6 - 310878143/106496*x^4 + 8028507297/79659008*x^2 - 131072/230945

20:

34461632205/131072*x^19 - 156402792315/131072*x^17 + 74513229329/32768*x^15 - 77414998155/32768*x^13 + 95261730395/65536*x^11 - 105848501135/196608*x^9 + 26743310835/229376*x^7 - 5794847907/425984*x^5 + 20766263345/28966912*x^3 - 66586053015/6054084608*x

21:

67282234305/131072*x^20 - 322188910615/131072*x^18 + 163522338309/32768*x^16 - 1282995547185/229376*x^14 + 742558730485/196608*x^12 - 103075775465/65536*x^10 + 13013113305/32768*x^8 - 1878498619/32768*x^6 + 121075899765/28966912*x^4 - 64646576905/550371328*x^2 + 524288/969969

22:

263012370465/262144*x^21 - 662628065125/131072*x^19 + 2855481893433/262144*x^17 - 3005556965469/229376*x^15 + 3789483198035/393216*x^13 - 3216377341645/720896*x^11 + 169097022925/131072*x^9 - 7318426051/32768*x^7 + 94274870637/4456448*x^5 - 502979176435/550371328*x^3 + 267102370903/23115595776*x

23:

514589420475/262144*x^22 - 1360803134145/131072*x^20 + 6203173794415/262144*x^18 - 6975326241117/229376*x^16 + 22208667128085/917504*x^14 - 26691745821715/2162688*x^12 + 528030441395/131072*x^10 - 26765985435/32768*x^8 + 428819002031/4456448*x^6 - 244962505155/42336256*x^4 + 1039899184795/7705198592*x^2 - 1048576/2028117

24:

8061900920775/2097152*x^23 - 44654926376775/2097152*x^21 + 107343353111085/2097152*x^19 - 1027485083834235/14680064*x^17 + 440106198614381/7340032*x^15 - 385307039593785/11534336*x^13 + 140315031972295/11534336*x^11 - 8921236789055/3145728*x^9 + 14348843260995/35651584*x^7 - 21408197984313/677380096*x^5 + 5406014269165/4741660672*x^3 - 17135157314895/1417756540928*x

25:

15801325804719/2097152*x^24 - 91475702447727/2097152*x^22 + 1156474541190969/10485760*x^20 - 335336198214693/2097152*x^18 + 1076890171411301/7340032*x^16 - 7162600782207351/80740352*x^14 + 411679201272879/11534336*x^12 - 148257227381443/15728640*x^10 + 56080269258699/35651584*x^8 - 104533336825617/677380096*x^6 + 5276308074547/677380096*x^4 - 16713284641491/109058195456*x^2 + 8388608/16900975

26:

61989816618513/4194304*x^25 - 93592468227951/1048576*x^23 + 2483156438378319/10485760*x^21 - 380494806475509/1048576*x^19 + 1489339830547347/4194304*x^17 - 9352343163683263/40370176*x^15 + 15314157627380727/149946368*x^13 - 870451548401283/28835840*x^11 + 73137250983157/12582912*x^9 - 233594244751521/338690048*x^7 + 30934486977297/677380096*x^5 - 10881416817269/7789871104*x^3 + 68678549172039/5452909772800*x

27:

121683714103007/4194304*x^26 - 286989891752375/1572864*x^24 + 5314782687120571/10485760*x^22 - 2145653569472083/2621440*x^20 + 32124164442057809/37748736*x^18 - 1718702333828119/2883584*x^16 + 300093020153493865/1049624576*x^14 - 4049615563778063/43253760*x^12 + 429631150101317/20971520*x^10 - 28575513593387/9961472*x^8 + 484154951918087/2032140288*x^6 - 39895022326015/3894935552*x^4 + 134218164762203/778987110400*x^2 - 16777216/35102025

28:

956086325095055/16777216*x^27 - 18756675359592079/50331648*x^25 + 9073800406090805/8388608*x^23 - 77012715588581839/41943040*x^21 + 305238570886767749/150994944*x^19 - 25386908875861127/16777216*x^17 + 470825734884649249/599785472*x^15 - 508118124328412989/1799356416*x^13 + 63992054323710559/922746880*x^11 - 3582813699887161/318767104*x^9 + 3792399412073011/3347054592*x^7 - 3998174474946391/62318968832*x^5 + 1050337841958499/623189688320*x^3 - 1100753944411999/84130607923200*x

29:

1879204156221315/16777216*x^28 - 12758807165923665/16777216*x^26 + 57937632446434195/25165824*x^24 - 34379175513839095/8388608*x^22 + 1198799345827030871/251658240*x^20 - 573151402442318371/150994944*x^18 + 115507176428338471/54525952*x^16 - 498473540851577539/599785472*x^14 + 125514154342377893/553648128*x^12 - 3512444631367411/83886080*x^10 + 5574772395112029/1115684864*x^8 - 3916574018979361/10997465088*x^6 + 8227550241799367/623189688320*x^4 - 1077284379476749/5608707194880*x^2 + 67108864/145422675

30:

7391536347803839/33554432*x^29 - 78049612621725283/50331648*x^27 + 820064912678600589/167772160*x^25 - 76383718903104665/8388608*x^23 + 3365174956648322455/301989888*x^21 - 474087236664589429/50331648*x^19 + 2455370552219168269/436207616*x^17 - 1957449328268086501/817889280*x^15 + 3449134640059994459/4798283776*x^13 - 27568974274300373/184549376*x^11 + 14580361302108559/704643072*x^9 - 46078616418790827/25660751872*x^7 + 32252863960796393/366582169600*x^5 - 33769795514029643/16826121584640*x^3 + 4409609607796471/325305017303040*x

31:

14544636039226909/33554432*x^30 - 159037249805972923/50331648*x^28 + 1737148445908765659/167772160*x^26 - 168993005727539565/8388608*x^24 + 7819521997112706205/301989888*x^22 - 1164907016500069045/50331648*x^20 + 19301683086810018377/1308622848*x^18 - 5528472207003406081/817889280*x^16 + 967832442369613699/436207616*x^14 - 94737063682454893/184549376*x^12 + 57243164081716441/704643072*x^10 - 11302866815235213/1350565888*x^8 + 189806410341546483/366582169600*x^6 - 16554598144617259/989771857920*x^4 + 69143903431133161/325305017303040*x^2 - 134217728/300540195

32:

916312070471295267/1073741824*x^31 - 6908702118632781775/1073741824*x^29 + 352516436189491547281/16106127360*x^27 - 238353199509736470159/5368709120*x^25 + 64208235123773921895/1073741824*x^23 - 544655275893124095115/9663676416*x^21 + 1598120347220690237525/41875931136*x^19 - 1309365766370543303303/69793218560*x^17 + 1400049091449770397803/209379655680*x^15 - 23827498559429176633/13958643712*x^13 + 3598300819703265679/11811160064*x^11 - 18940764920721223595/518617300992*x^9 + 11923596729414398037/4321810841600*x^7 - 4158338218477272679/35191888281600*x^5 + 4340370004865262679/1837016568299520*x^3 - 4521361609616749879/322702577164615680*x

33:

1804857108504066435/1073741824*x^32 - 14059374091372702127/1073741824*x^30 + 49580019778444673343/1073741824*x^28 - 523387817477074139247/5368709120*x^26 + 147446382863797181255/1073741824*x^24 - 1607769757707276938385/11811160064*x^22 + 314434281996836139145/3221225472*x^20 - 6439108033576786149715/125627793408*x^18 + 1376691265292999782553/69793218560*x^16 - 76965085677253966937/13958643712*x^14 + 3536455038862643719/3221225472*x^12 - 3721992381384509599/24696061952*x^10 + 11711802116921343537/864362168320*x^8 - 4083144864929442679/5556613939200*x^6 + 4260386341149327679/204112952033280*x^4 - 4436350401324384679/18982504539095040*x^2 + 4294967296/9917826435

34:

7113260368810144185/2147483648*x^33 - 3574324861939425685/134217728*x^31 + 52203654339274631067/536870912*x^29 - 57271565542718197573/268435456*x^27 + 421030837082304112027/1342177280*x^25 - 15053787122738682045/46137344*x^23 + 132638749650060717495/536870912*x^21 - 333503479991229075535/2415919104*x^19 + 13544738506485891444005/237296943104*x^17 - 151411190102307181249/8724152320*x^15 + 79987863179894870087/20937965568*x^13 - 3659237103766819669/6174015488*x^11 + 959260930920901951/15435038720*x^9 - 2005993812605876777/486203719680*x^7 + 4184798064526141429/26856967372800*x^5 - 4356131491538696929/1581875378257920*x^3 + 18106386699333133591/1252845299580272640*x

35:

14023284727082855679/2147483648*x^34 - 29062749506852874813/536870912*x^32 + 274361536354520391413/1342177280*x^30 - 249895325175104877483/536870912*x^28 + 382856276654104159011/536870912*x^26 - 4586500613244517506175/5905580032*x^24 + 1828641383791711321875/2952790016*x^22 - 197014694033740493715/536870912*x^20 + 26666125397659606552225/164282499072*x^18 - 372532799875530161875/6979321856*x^16 + 44973247280727134999/3489660928*x^14 - 3599600100813362779/1610612736*x^12 + 83017909907407464913/308700774400*x^10 - 3944509263193754479/185220464640*x^8 + 8226410336477504333/8057090211840*x^6 - 4280255259575993779/166513197711360*x^4 + 35569876135955255357/139205033286696960*x^2 - 8589934592/20419054425

36:

110628135069209194801/8589934592*x^35 - 2834261657618190497789/25769803776*x^33 + 2303409198094416308897/5368709120*x^31 - 1087205332739281021591/1073741824*x^29 + 31191805230431669119891/19327352832*x^27 - 43386699161226175132287/23622320128*x^25 + 18172669044905127959325/11811160064*x^23 - 1035022195230427787005/1073741824*x^21 + 33173249205098237839275/73014444032*x^19 - 11738119885696888985525/73014444032*x^17 + 1770952406121309040805/41875931136*x^15 - 113370483413834338853/13958643712*x^13 + 59410321425207074539/53687091200*x^11 - 682949748673535387429/6667936727040*x^9 + 64724131303752271789/10742786949120*x^7 - 67333294635114902689/333026395422720*x^5 + 279690284203368590731/87918968391598080*x^3 - 290000525134771195231/19488704660137574400*x

37:

218266320541953276229/8589934592*x^36 - 5755652973195343243025/25769803776*x^34 + 4827336791442146634653/5368709120*x^32 - 11792919126618608357951/5368709120*x^30 + 70293679006714804512703/19327352832*x^28 - 101991172459178696554875/23622320128*x^26 + 44779267146802151391225/11811160064*x^24 - 29579653568541620089595/11811160064*x^22 + 91519760377334184310695/73014444032*x^20 - 34690290757254772994025/73014444032*x^18 + 436063639126756912565/3221225472*x^16 - 398707916152286869625/13958643712*x^14 + 233958745540977075031/53687091200*x^12 - 672209508796522491179/1449551462400*x^10 + 350297214235742572777/10742786949120*x^8 - 463681867271403090373/333026395422720*x^6 + 550143149842521064187/17583793678319616*x^4 - 285124059829378071481/1025721297901977600*x^2 + 34359738368/83945001525

38:

861577581086657669325/17179869184*x^37 - 3894330666511692665349/8589934592*x^35 + 96984680329002412468067/51539607552*x^33 - 25518291442786553428303/5368709120*x^31 + 105194818896631078926883/12884901888*x^29 - 2145765951019244987008849/212600881152*x^27 + 438010417946615325714651/47244640256*x^25 - 76075336113213453716175/11811160064*x^23 + 498320940605253691337195/146028888064*x^21 - 1914591013691371387302525/1387274436608*x^19 + 61874664330856402306575/146028888064*x^17 - 314239919353750972015/3221225472*x^15 + 921779307827465631649/55834574848*x^13 - 962865125118605739499/483183820800*x^11 + 2759053362803102846591/16814796963840*x^9 - 2868808385546291776841/333026395422720*x^7 + 15158163213802592430461/58612645594398720*x^5 - 2243957007920907321473/615432778741186560*x^3 + 1161071500825355202799/75903376044746342400*x

39:

1701063429324939500975/17179869184*x^38 - 23704429346307273825275/25769803776*x^36 + 67566239412786596978727/17179869184*x^34 - 11017165329073672749211/1073741824*x^32 + 3529044338226234676775047/193273528320*x^30 - 151209585122990405816621/6442450944*x^28 + 13826043762928987006066209/614180323328*x^26 - 193831103392203017666975/11811160064*x^24 + 14742191957304435107323875/1606317768704*x^22 - 5474378847350535330631185/1387274436608*x^20 + 189764391051982198080825/146028888064*x^18 - 348434975017569124125/1073741824*x^16 + 259527830291641928639/4294967296*x^14 - 474321555720451697237/57982058496*x^12 + 21741754718519061252103/28024661606400*x^10 - 706296949896063787679/14479408496640*x^8 + 29848132138279325073847/15985266980290560*x^6 - 1104365175702640878049/29306322797199360*x^4 + 4570109547711607628821/15180675208949268480*x^2 - 68719476736/172308161025

40:

26876802183334044115405/137438953472*x^39 - 769220882740737642340895/412316860416*x^37 + 1128298141119610637843331/137438953472*x^35 - 3037299564045230743417515/137438953472*x^33 + 12584887356221794419142645/309237645312*x^31 - 5598742477280219239434911/103079215104*x^29 + 72764386604703176837212409/1340029796352*x^27 - 78333034971854151285635489/1889785610240*x^25 + 313627349474306906397835665/12850542149632*x^23 - 246841534973975513504117745/22196390985728*x^21 + 86834890007390449505783055/22196390985728*x^19 - 1231339456098432471911085/1168231104512*x^17 + 36843174399434762907261/171798691840*x^15 - 29921385947571216865403/927712935936*x^13 + 31164796286250336173303/8967891714048*x^11 - 356293602742223783579173/1390023215677440*x^9 + 67203580318841085974611/5560092862709760*x^7 - 69605278463574396683611/213136893070540800*x^5 + 71992151311117995598111/17349343095942021120*x^3 - 74370259631681630257111/4736370665192171765760*x

41:

53098072606098965203605/137438953472*x^40 - 519836686131314560573565/137438953472*x^38 + 7057250806675659518148913/412316860416*x^36 - 6527864909750131737805155/137438953472*x^34 + 3106486776273425631494485/34359738368*x^32 - 38691542917696885231616741/309237645312*x^30 + 174438513736688719517280289/1340029796352*x^28 - 231943139348223504464479167/2233382993920*x^26 + 825349876269452067002230265/12850542149632*x^24 - 7550447189322515804504830995/244160300843008*x^22 + 257005574461048024916399655/22196390985728*x^20 - 3913734172697739844162485/1168231104512*x^18 + 127177584547967272924101/171798691840*x^16 - 4214952769171684579529/34359738368*x^14 + 399425696871473541239789/26903675142144*x^12 - 351197186261765580458953/278004643135488*x^10 + 132457001270159900347097/1853364287569920*x^8 - 68580163401797983576111/27800464313548800*x^6 + 70915147709177591209861/1577213008722001920*x^4 - 73239208113364779626611/225541460247246274560*x^2 + 549755813888/1412926920405

42:

209863810776486386280915/274877906944*x^41 - 526766593314473861146875/68719476736*x^39 + 4899125977352777645574099/137438953472*x^37 - 7000869182085297266540529/68719476736*x^35 + 165126127406947395733755305/824633720832*x^33 - 4930394766896739478054283/17179869184*x^31 + 415774437631798108917930275/1340029796352*x^29 - 2595376462559707280831566301/10050223472640*x^27 + 1955482464946265345013426111/11682311045120*x^25 - 10369081213741117760212082805/122080150421504*x^23 + 748856324147618621945952355/22196390985728*x^21 - 115859396198182668618825525/11098195492864*x^19 + 29110463433073442901599307/11682311045120*x^17 - 116420555342252656417937/257698037760*x^15 + 60607460962958095922281/996432412672*x^13 - 818569315947032386361443/139002321567744*x^11 + 2611826976171271876954615/6672111435251712*x^9 - 77257490189331965323859/4633410718924800*x^7 + 139773285433282670932847/342872393200435200*x^5 - 144320360137127170009097/30755653670079037440*x^3 + 297704555850584374843519/18494399740274194513920*x

43:

414847067813984717066925/274877906944*x^42 - 66700901099503425097035/4294967296*x^40 + 10192066253313983986462515/137438953472*x^38 - 3746945558860348164435973/17179869184*x^36 + 364668436903431670588281395/824633720832*x^34 - 351976582476718600215115/536870912*x^32 + 328547969067625367210684653/446676598784*x^30 - 320412943191432634323984217/502511173632*x^28 + 3862158317023460376183841797/8933531975680*x^26 - 159974432320230759031701495/693637218304*x^24 + 23658126628594629658341124505/244160300843008*x^22 - 88634684537950786952584665/2774548873216*x^20 + 19154222066717643633414813/2336462209024*x^18 - 13014487667596957841383/8053063680*x^16 + 239196433830291859796749/996432412672*x^14 - 100939126664475136492091/3861175599104*x^12 + 13395629226195763131250393/6672111435251712*x^10 - 1190399523224418987731/11583526797312*x^8 + 1102447765208380988877151/342872393200435200*x^6 - 17782289328986920286059/334300583370424320*x^4 + 586766849872574995551263/1681309067297654046720*x^2 - 1099511627776/2893136075115

44:

3281063172710606398620225/1099511627776*x^43 - 34583160107765816868215475/1099511627776*x^41 + 84728335551214668989170005/549755813888*x^39 - 256230593272088341927920315/549755813888*x^37 + 1070866742683471779718648313/1099511627776*x^35 - 54135758148027598000497725795/36283883716608*x^33 + 3099623150752178334890386655/1786706395136*x^31 - 2794366776926657268079858793/1786706395136*x^29 + 71215990279089250530145580629/64321430224896*x^27 - 32359985933991032718280955901/52226802319360*x^25 + 24380148769169164526065429365/88785563942912*x^23 - 8535505670614668042061352555/88785563942912*x^21 + 4689976838397425811679334199/177571127885824*x^19 - 52623316130440634545709743/9345848836096*x^17 + 1888727934708805649345117/2061584302080*x^15 - 13731216147463439714808719/123557619171328*x^13 + 9612615524557933460837719/988460953370624*x^11 - 15632896443926070456565847/26688445741006848*x^9 + 1242713779474848944337569/54859582912069632*x^7 - 8978247470802712780299083/17829364446422630400*x^5 + 4628036867603805836928979/877204730763993415680*x^3 - 4766527771440898294508479/289185159575196496035840*x

45:

6489213830472088210604445/1099511627776*x^44 - 70020318522809311366110135/1099511627776*x^42 + 175921807534455751781836257/549755813888*x^40 - 546629476928869413626718015/549755813888*x^38 + 2352411973168438418355808885/1099511627776*x^36 - 40919115326070369898542299085/12094627905536*x^34 + 7275713229826105108265425155/1786706395136*x^32 - 6811624126719434699737509589/1786706395136*x^30 + 20105651434303930601717619225/7146825580544*x^28 - 223802493945573672680043701985/135789686030336*x^26 + 4014129729782477487171720305/5222680231936*x^24 - 25293111794271545117264852655/88785563942912*x^22 + 74111467808471145921993240039/887855639429120*x^20 - 1610923643870304350809834843/84112639524864*x^18 + 466210333743549386543783/137438953472*x^16 - 1936501134178021855685087/4260607557632*x^14 + 132833406962286934364098001/2965382860111872*x^12 - 15427812869629113505079957/4942304766853120*x^10 + 7970260902685752941897341/54859582912069632*x^8 - 8857238697861912373326773/2139523733570715648*x^6 + 18259016873957426578028941/292401576921331138560*x^4 - 4700359895163176342553829/12573267807617238958080*x^2 + 4398046511104/11835556670925

46:

25674715590128696833261065/2199023255552*x^45 - 141728191920600535266389835/1099511627776*x^43 + 1459739616353061242754951963/2199023255552*x^41 - 1164262119572429929229217165/549755813888*x^39 + 10310010563165906384517018995/2199023255552*x^37 - 92472142006537438950610437645/12094627905536*x^35 + 230168509643186726812998599235/24189255811072*x^33 - 16507409506854708320569147219/1786706395136*x^31 + 101417404363845912739594692705/14293651161088*x^29 - 589816217946971369874952326965/135789686030336*x^27 + 44426586270710008570860073853/20890720927744*x^25 - 99964113761509500274824879795/120121645334528*x^23 + 460222657493368473818651891013/1775711278858240*x^21 - 101841904343537253832181191177/1598140150972416*x^19 + 228389511785865746021298593/18691697672192*x^17 - 7649419589959106831661713/4260607557632*x^15 + 40356152133176350703337833/204509162766336*x^13 - 77538466612212703427190497/4942304766853120*x^11 + 62937294585726522752028953/73146110549426176*x^9 - 4994935459002999871714331/164578748736208896*x^7 + 72065366686804590004011799/116960630768532455424*x^5 - 74191663697050082869382389/12573267807617238958080*x^3 + 19078010842169270446073191/1131594102685551506227200*x

47:

50803160635786570329644235/2199023255552*x^46 - 286792035847182251860894875/1099511627776*x^44 + 3025481992019854302593460957/2199023255552*x^42 - 2475931495038743574474077961/549755813888*x^40 + 22540655449560452465263805985/2199023255552*x^38 - 208302758830451311959464042485/12094627905536*x^36 + 535536808655634885622867260525/24189255811072*x^34 - 3060346796578541258253912027/137438953472*x^32 + 254010559244051528814589573411/14293651161088*x^30 - 1540962715375531334984505651285/135789686030336*x^28 + 1580884515835769749973060635551/271579372060672*x^26 - 288161953913409461850774338375/120121645334528*x^24 + 82691326655335531380498485991/104453604638720*x^22 - 553465483634410478323896845911/2663566918287360*x^20 + 7220574395762502759453866551/168225279049728*x^18 - 944551789831501963319131/137438953472*x^16 + 56942671295393424629525465/68169720922112*x^14 - 38286805419026661159155311/511272906915840*x^12 + 1740045042352773094701822059/365730552747130880*x^10 - 3698404260657004477662567/18286527637356544*x^8 + 142267178465284365884609603/26990914792738258944*x^6 - 183047489789418925439122585/2514653561523447791616*x^4 + 150593935987924058224522403/377198034228517168742400*x^2 - 8796093022208/24185702762325

48:

1608766753466574727105400775/35184372088832*x^47 - 18565732814566892201577765435/35184372088832*x^45 + 100247746427474203020473145135/35184372088832*x^43 - 336487243361105478903210789147/35184372088832*x^41 + 786747848729286603631718131865/35184372088832*x^39 - 14969266546314174291031155630855/387028092977152*x^37 + 19856126056966611131843256962835/387028092977152*x^35 - 20662306130674328666072114418495/387028092977152*x^33 + 777892028275955527331282277651/17592186044416*x^31 - 6725415052952023749535335689995/228698418577408*x^29 + 205626753491413435015686279112055/13035809858912256*x^27 - 52508894081911038256131329472531/7687785301409792*x^25 + 18309787711848471088314971926509/7687785301409792*x^23 - 10002931564535467393527665726819/15041319067975680*x^21 + 2511808421404868815671029010577/17046828277039104*x^19 - 7645972059519565143028068797/299067162755072*x^17 + 360061848539084634748855721/105553116266496*x^15 - 372404195131366845233864441/1090715534753792*x^13 + 4999805355181485362257949933/201782373929451520*x^11 - 4363446464578808270857136251/3511013306372456448*x^9 + 1926531276748042139637480579/47983848520423571456*x^7 - 4626234080359530057219003511/6189916459134640717824*x^5 + 4756666789267736095086351871/724220225718752963985408*x^3 - 4886765910685128306769896271/283652921739844910894284800*x

49:

3184701532372607112841303575/35184372088832*x^48 - 37548834905740257746249184075/35184372088832*x^46 + 207439672867883053711106726655/35184372088832*x^44 - 4994690396556925651143317794605/246290604621824*x^42 + 1712665364887751445164510740553/35184372088832*x^40 - 33519230861844114462378994472535/387028092977152*x^38 + 45836678879374173366500667237795/387028092977152*x^36 - 49296652320876274954435860299535/387028092977152*x^34 + 1923638723692780482486661559715/17592186044416*x^32 - 1330369497073434659080898515399/17592186044416*x^30 + 203357528429455275098723091090155/4802666790125568*x^28 - 1921193051530874614377668357780295/99941208918327296*x^26 + 54311896540230521681063898864477/7687785301409792*x^24 - 6293252292251508286017098410903/3008263813595136*x^22 + 2483013286006981837101986345077/5013773022658560*x^20 - 83131528870110084960027711367/897201488265216*x^18 + 1423386867115095876457866329/105553116266496*x^16 - 367996214205551770060647041/246290604621824*x^14 + 4939944331880184282196160633/40356474785890304*x^12 - 4310580797256517397296769251/605347121788354560*x^10 + 13320305046588390112559005203/47983848520423571456*x^8 - 13706270791433501464000175853/2063305486378213572608*x^6 + 4696774218850702710351344971/55709248132211766460416*x^4 - 4824371434086991225656359671/11346116869593796435771392*x^2 + 140737488355328/395033145117975

50:

12611418068195524166851562157/70368744177664*x^49 - 18980821132940738392534169307/8796093022208*x^47 + 536156213045855000881921911129/43980465111040*x^45 - 661074157854699113874013438059/15393162788864*x^43 + 3720834920971631230750877839659/35184372088832*x^41 - 1701118611721660368696825693571/8796093022208*x^39 + 6589156587284297129430115853067/24189255811072*x^37 - 73166484456976439574368372010411/241892558110720*x^35 + 208085525624236719819739299342621/774056185954304*x^33 - 849116447978959260441208923147/4398046511104*x^31 + 3468313769744064701672493133709/30786325577728*x^29 - 26392621298996470451738353250659/493130965057536*x^27 + 3974906783748231260487880565868621/192194632535244800*x^25 - 12446862139101033398448797611787/1921946325352448*x^23 + 613799296303155753951072782551/376032976699392*x^21 - 41095443890222888496719492147/125344325566464*x^19 + 61913011771379446828384550107/1196268651020288*x^17 - 5819902241698872635671742087/923589767331840*x^15 + 187777950746662311991357441/325455441821696*x^13 - 96809907178339464215358233/2522279674118144*x^11 + 8773996238741777710056559499/4963846398664507392*x^9 - 27080611042300207165760944947/515826371594553393152*x^7 + 72486975610670495473332283/80597870561648967680*x^5 - 2382200371462967811731501381/327291832776744127954944*x^3 + 19557358731379205845463318623/1111919453220192050705596416*x

Definition

For $n\geq 0$ the secondary polynomial of the Legendre polynomial $P_n$ is $q_n(x)=\int_{-1}^{1}\frac{P_n(t)-P_n(x)}{t-x}\,\mathrm{d}t$ [2], the integral taken against the density $1$ on $[-1,1]$.

Parameters

$n$

— integer ($n\geq 0$)

Formulas

(1)

$q_0=0$, $q_1=2$, and $(n+1)\,q_{n+1}(x)=(2n+1)\,x\,q_n(x)-n\,q_{n-1}(x)$ for $n\geq 1$: Bonnet's recurrence for $P_n$ with the other pair of starting values.

(2)

$q_n(x)=2\sum_{k=1}^{n}\frac{1}{k}\,P_{k-1}(x)\,P_{n-k}(x)$ [4].

(3)

If $P_n(x)=\sum_{j=0}^{n}c_j x^j$ then $q_n(x)=\sum_{j=1}^{n}c_j\sum_{i=0}^{j-1}m_i\,x^{j-1-i}$ with $m_i=\int_{-1}^{1}t^i\,\mathrm{d}t=\frac{1+(-1)^i}{i+1}$.

(4)

$Q_n(x)=\frac12P_n(x)\ln\frac{1+x}{1-x}-\frac12q_n(x)$ for $-1<x<1$, and $Q_n(x)=\frac12P_n(x)\ln\frac{x+1}{x-1}-\frac12q_n(x)$ for $x>1$, where $Q_n$ is the Legendre function of the second kind [4], [1].

(5)

$\frac{t^{n}\,q_n(1/t)}{2\,t^{n}\,P_n(1/t)}$ is the $[n-1/n]$ Padé approximant of $\operatorname{artanh}t=\frac12\ln\frac{1+t}{1-t}$ at $t=0$: $t^{n}P_n(1/t)\operatorname{artanh}t-\frac12t^{n}q_n(1/t)=O(t^{2n+1})$.

(6)

$q_n(1)=2H_n=2\bigl(1+\frac12+\cdots+\frac1n\bigr)$; $q_n(-x)=(-1)^{n-1}q_n(x)$; $q_n(0)=(-1)^{(n-1)/2}\,\frac{2\,(n-1)!!}{n!!}$ for odd $n$ and $0$ for even $n$; the leading coefficient is $\binom{2n}{n}/2^{n-1}$, twice that of $P_n$.

(7)

At a root $x_k$ of $P_n$, $\dfrac{q_n(x_k)}{P_n'(x_k)}=\dfrac{2}{(1-x_k^2)\,P_n'(x_k)^2}=w_k$, the Gauss–Legendre weight.

Comments

(8)

$P_n$ is the Legendre polynomial of degree $n$ with $P_n(1)=1$, orthogonal on $[-1,1]$ for the density $\rho(t)=1$. The quotient $(P_n(t)-P_n(x))/(t-x)$ is a polynomial in $t$ and $x$, so $q_n$ is a combination of the moments $\int_{-1}^{1}t^i\,\mathrm{d}t$ and is rational; it has degree $n-1$ because $P_n$ has degree $n$.

(9)

[2] defines the secondary polynomials for "a density" and fixes none. Taking $\rho=1$, the inner product the table of Legendre polynomials states, gives $q_1=2$, the length of the interval; the probability density $\frac12$ on $[-1,1]$ would halve every entry. Under $\rho=1$ the values at the roots of $P_n$, divided by $P_n'$ there, are the Gauss–Legendre weights, and $q_n$ is twice the polynomial part $W_{n-1}$ of the Legendre function of the second kind of [4].

(10)

$q_n=2W_{n-1}$, where $W_{n-1}$ is the polynomial part of the Legendre function of the second kind, $Q_n(x)=\frac12P_n(x)\ln\frac{1+x}{1-x}-W_{n-1}(x)$ [4]. Anyone computing $Q_n$ by hand from $Q_0=\operatorname{artanh}x$ and Bonnet's recurrence produces $W_{n-1}$, and $W_{n-1}/P_n$ is the $n$-th convergent of Gauss's continued fraction for $\operatorname{artanh}$.

(11)

At the roots $x_1<\cdots<x_n$ of $P_n$, $q_n(x_k)/P_n'(x_k)$ is the weight $w_k$ of the $n$-point Gauss–Legendre quadrature rule, whose nodes and weights are in the table of Gauss–Legendre nodes and weights.

(12)

The same construction for the other classical families: for the Chebyshev polynomials $T_n$ with the density $(1-x^2)^{-1/2}$ it gives $\pi\,U_{n-1}$, with $U_{n-1}$ the Chebyshev polynomial of the second kind; for the Hermite polynomials $H_n$ with $e^{-x^2}$ it gives $\sqrt{\pi}$ times a polynomial with integer coefficients ($2$, $4x$, $8x^2-8$, $16x^3-40x$); for the Laguerre polynomials $L_n$ with $e^{-x}$ the result is rational as it stands ($-1$, $\frac12x-\frac32$, $-\frac16x^2+\frac43x-\frac{11}{6}$).

(13)

For $n\geq 1$ the $q_n$ are themselves orthogonal on $[-1,1]$, for the secondary measure $d\mu=\dfrac{du}{\pi^2+\ln^2\frac{1+u}{1-u}}$ of the density $1$ [3] (given there on $[0,1]$ as $1/(\pi^2+\ln^2\frac{x}{1-x})$), and $\int_{-1}^{1}q_n^2\,\mathrm{d}\mu=\frac{2}{2n+1}=\int_{-1}^{1}P_n^2\,\mathrm{d}t$.

(14)

Up to the index and a factor $2$ these are the associated polynomials of order one, or numerator polynomials, of the Legendre family: $q_{n+1}/2$ has degree $n$, starts at $1$ and satisfies Bonnet's recurrence shifted by one step. They are not the associated Legendre functions $P_n^m$, and not the Stieltjes polynomials of Gauss–Kronrod quadrature.

(15)

$q_2=3x$ is also the Gegenbauer polynomial $C_1^{(3/2)}$.

Programs

(P1)

Sage

R.<x> = QQ[]
P = R(legendre_P(8, x))
m = lambda i: 2/(i + 1) if i % 2 == 0 else 0            # moments of the density 1 on [-1, 1]
q = sum(c * sum(m(i) * x^(j - 1 - i) for i in range(j)) for j, c in enumerate(P.list()))
q          # 6435/64*x^7 - 9867/64*x^5 + 4213/64*x^3 - 15159/2240*x

References

[1]

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Dover, 1972, §8.6.19.

Links

[2]

Wikipedia: Secondary polynomials

[3]

Wikipedia: Secondary measure

[4]

DLMF 14.7(i): Legendre function of the second kind, W_{n-1}

[5]

Wikipedia: Legendre function (Legendre functions of the second kind)

[6]

MathWorld: Legendre Function of the Second Kind

Similar tables

Legendre polynomials — the $P_n$ these are the secondary polynomials of, and the recurrence they share

Nodes and weights of Gauss–Legendre quadrature — the weights are $q_n(x_k)/P_n'(x_k)$ at the nodes

Chebyshev polynomials of the second kind — the same construction for $T_n$ gives $\pi\,U_{n-1}$

Data properties

Entries are of type: rational polynomial

How well the digits are known: exact

Table is complete: no (every $n\leq 50$ is here, the range of the table of Legendre polynomials)

How they were obtained:

Every value is exact, a polynomial with rational coefficients, so there is no precision to choose. Each entry is built by the recurrence and, before it is written, must equal the defining integral evaluated exactly from the moments and the sum of [4], and must have degree $n-1$, the parity of $n-1$ and $q_n(1)=2H_n$.

Outside the generator the same polynomials were compared with a computation in plain Python fractions, with the Legendre function of the second kind in ball arithmetic at rational points, with the Padé property for $\operatorname{artanh}$, and with every Gauss–Legendre weight for $n\leq 30$.