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Secondary polynomials of the Jacobi polynomials $q_n^{(\alpha,\beta)}$
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Polynomials
$\alpha$
$\beta$
$n$ 
$q_n^{(\alpha,\beta)}(x)$
-1/2
0
0:
0
-1/2
0
1:
3/4
-1/2
0
2:
35/32*x + 5/96
-1/2
0
3:
231/128*x^2 + 7/64*x - 301/640
-1/2
0
4:
6435/2048*x^3 + 429/2048*x^2 - 3311/2048*x - 3083/71680
-1/2
0
5:
46189/8192*x^4 + 2431/6144*x^3 - 88517/20480*x^2 - 12441/71680*x + 955823/2580480
-1/2
0
6:
676039/65536*x^5 + 146965/196608*x^4 - 5160571/491520*x^3 - 83317/163840*x^2 + 1186315/589824*x + 8413379/227082240
-1/2
0
7:
5014575/262144*x^6 + 185725/131072*x^5 - 19085101/786432*x^4 - 1804601/1376256*x^3 + 67969859/9175040*x^2 + 101044549/454164480*x - 3722627477/11808276480
-1/2
0
8:
300540195/8388608*x^7 + 22621305/8388608*x^6 - 456593769/8388608*x^5 - 185613565/58720256*x^4 + 4054042445/176160768*x^3 + 565851103/645922816*x^2 - 19658625291/8396996608*x - 12455945101/377864847360
-1/2
0
9:
2268783825/33554432*x^8 + 21607465/4194304*x^7 - 1003764965/8388608*x^6 - 30681333/4194304*x^5 + 22871354215/352321536*x^4 + 251421395/88080384*x^3 - 8379311999/763363328*x^2 - 3316358249/12595494912*x + 2390529851191/8564936540160
-1/2
0
10:
34461632205/268435456*x^9 + 2650894785/268435456*x^8 - 17403562099/67108864*x^7 - 1105619867/67108864*x^6 + 23096849987/134217728*x^5 + 3360676165/402653184*x^4 - 58467416915/1409286144*x^3 - 7946594829/6106906624*x^2 + 3276809735477/1245808951296*x + 7802532059081/260374070820864
-1/2
1/2
0:
0
-1/2
1/2
1:
1
-1/2
1/2
2:
3/2*x
-1/2
1/2
3:
5/2*x^2 - 5/8
-1/2
1/2
4:
35/8*x^3 - 35/16*x
-1/2
1/2
5:
63/8*x^4 - 189/32*x^2 + 63/128
-1/2
1/2
6:
231/16*x^5 - 231/16*x^3 + 693/256*x
-1/2
1/2
7:
429/16*x^6 - 2145/64*x^4 + 1287/128*x^2 - 429/1024
-1/2
1/2
8:
6435/128*x^7 - 19305/256*x^5 + 32175/1024*x^3 - 6435/2048*x
-1/2
1/2
9:
12155/128*x^8 - 85085/512*x^6 + 182325/2048*x^4 - 60775/4096*x^2 + 12155/32768
-1/2
1/2
10:
46189/256*x^9 - 46189/128*x^7 + 969969/4096*x^5 - 230945/4096*x^3 + 230945/65536*x
-1/2
1
0:
0
-1/2
1
1:
5/4
-1/2
1
2:
63/32*x - 21/160
-1/2
1
3:
429/128*x^2 - 99/320*x - 3441/4480
-1/2
1
4:
12155/2048*x^3 - 1287/2048*x^2 - 39897/14336*x + 4895/43008
-1/2
1
5:
88179/8192*x^4 - 12597/10240*x^3 - 157131/20480*x^2 + 35269/71680*x + 1898819/3153920
-1/2
1
6:
1300075/65536*x^5 - 156009/65536*x^4 - 3115981/163840*x^3 + 739993/491520*x^2 + 12248177/3604480*x - 14077813/140574720
-1/2
1
7:
9694845/262144*x^6 - 601749/131072*x^5 - 81830435/1835008*x^4 + 1834963/458752*x^3 + 185269893/14417920*x^2 - 1242759619/1968046080*x - 402561241/787218432
-1/2
1
8:
583401555/8388608*x^7 - 74251107/8388608*x^6 - 5936324679/58720256*x^5 + 578760615/58720256*x^4 + 2389356495/58720256*x^3 - 21608110449/8396996608*x^2 - 164364365007/41984983040*x + 64350310911/713744711680
-1/2
1
9:
4418157975/33554432*x^8 - 71645805/4194304*x^7 - 1879849455/8388608*x^6 + 97285285/4194304*x^5 + 13677785475/117440512*x^4 - 3283171225/381681664*x^3 - 2047369955/109051904*x^2 + 4106018535/5490343936*x + 4906209165197/10848919617536
-1/2
1
10:
67282234305/268435456*x^9 - 8861562567/268435456*x^8 - 229668474849/469762048*x^7 + 3552494693/67108864*x^6 + 41916145599/134217728*x^5 - 3443237425/134217728*x^4 - 33920006861/469762048*x^3 + 394781200851/103817412608*x^2 + 34478278171801/7890123358208*x - 651553617151/7890123358208
-1/2
3/2
0:
0
-1/2
3/2
1:
3/2
-1/2
3/2
2:
5/2*x - 1/3
-1/2
3/2
3:
35/8*x^2 - 5/6*x - 85/96
-1/2
3/2
4:
63/8*x^3 - 7/4*x^2 - 161/48*x + 7/24
-1/2
3/2
5:
231/16*x^4 - 7/2*x^3 - 609/64*x^2 + 21/16*x + 175/256
-1/2
3/2
6:
429/16*x^5 - 55/8*x^4 - 385/16*x^3 + 33/8*x^2 + 1023/256*x - 33/128
-1/2
3/2
7:
6435/128*x^6 - 429/32*x^5 - 29315/512*x^4 + 715/64*x^3 + 15873/1024*x^2 - 429/256*x - 4719/8192
-1/2
3/2
8:
12155/128*x^7 - 5005/192*x^6 - 33605/256*x^5 + 3575/128*x^4 + 153725/3072*x^3 - 3575/512*x^2 - 9295/2048*x + 715/3072
-1/2
3/2
9:
46189/256*x^8 - 2431/48*x^7 - 901901/3072*x^6 + 17017/256*x^5 + 595595/4096*x^4 - 12155/512*x^3 - 182325/8192*x^2 + 12155/6144*x + 99671/196608
-1/2
3/2
10:
88179/256*x^9 - 12597/128*x^8 - 247741/384*x^7 + 29393/192*x^6 + 1616615/4096*x^5 - 146965/2048*x^4 - 356915/4096*x^3 + 20995/2048*x^2 + 986765/196608*x - 20995/98304
-1/2
2
0:
0
-1/2
2
1:
7/4
-1/2
2
2:
99/32*x - 135/224
-1/2
2
3:
715/128*x^2 - 715/448*x - 2585/2688
-1/2
2
4:
20995/2048*x^3 - 49725/14336*x^2 - 165035/43008*x + 249145/473088
-1/2
2
5:
156009/8192*x^4 - 14535/2048*x^3 - 46189/4096*x^2 + 55675/22528*x + 841847/1171456
-1/2
2
6:
2340135/65536*x^5 - 928625/65536*x^4 - 958341/32768*x^3 + 8645095/1081344*x^2 + 123390199/28114944*x - 13081993/28114944
-1/2
2
7:
17678835/262144*x^6 - 25662825/917504*x^5 - 130149525/1835008*x^4 + 111598875/5046272*x^3 + 4616433477/262406144*x^2 - 410096551/131203072*x - 2651592101/4460904448
-1/2
2
8:
1074687075/8388608*x^7 - 460580175/8388608*x^6 - 9690792975/58720256*x^5 + 3299398875/58720256*x^4 + 44470663875/763363328*x^3 - 112080413925/8396996608*x^2 - 693929578635/142748942336*x + 1139100471525/2712229904384
-1/2
2
9:
8205150525/33554432*x^8 - 3155827125/29360128*x^7 - 3133082425/8388608*x^6 + 568100575/4194304*x^5 + 20286721625/117440512*x^4 - 17655306575/381681664*x^3 - 318679262875/12977176576*x^2 + 452177607425/123283177472*x + 509757855805/986265419776
-1/2
2
10:
125788525005/268435456*x^9 - 394917462225/1879048192*x^8 - 389304663605/469762048*x^7 + 21151110125/67108864*x^6 + 63973905315/134217728*x^5 - 19055127585/134217728*x^4 - 786524019825/7985954816*x^3 + 38296341089505/1972530839552*x^2 + 41701786788325/7890123358208*x - 69930359980905/181472837238784
0
1/2
0:
0
0
1/2
1:
5/4
0
1/2
2:
63/32*x - 7/160
0
1/2
3:
429/128*x^2 - 33/320*x - 3849/4480
0
1/2
4:
12155/2048*x^3 - 429/2048*x^2 - 43329/14336*x + 8657/215040
0
1/2
5:
88179/8192*x^4 - 4199/10240*x^3 - 167739/20480*x^2 + 12311/71680*x + 6570889/9461760
0
1/2
6:
1300075/65536*x^5 - 52003/65536*x^4 - 3289109/163840*x^3 + 256139/491520*x^2 + 41268163/10813440*x - 35652769/984023040
0
1/2
7:
9694845/262144*x^6 - 200583/131072*x^5 - 85693515/1835008*x^4 + 631465/458752*x^3 + 286047831/20185088*x^2 - 29567743/131203072*x - 2355120517/3936092160
0
1/2
8:
583401555/8388608*x^7 - 24750369/8388608*x^6 - 6179950431/58720256*x^5 + 198321525/58720256*x^4 + 2600340095/58720256*x^3 - 1090965693/1199570944*x^2 - 187992829911/41984983040*x + 5436600131/164710318080
0
1/2
9:
4418157975/33554432*x^8 - 23881935/4194304*x^7 - 1948083555/8388608*x^6 + 33227939/4194304*x^5 + 14736261115/117440512*x^4 - 1151905435/381681664*x^3 - 16116736985/763363328*x^2 + 19299094357/71374471168*x + 28939651317329/54244598087680
0
1/2
10:
67282234305/268435456*x^9 - 2953854189/268435456*x^8 - 237141473481/469762048*x^7 + 1210245487/67108864*x^6 + 44803316463/134217728*x^5 - 1201230315/134217728*x^4 - 5378352499/67108864*x^3 + 20197342383/14831058944*x^2 + 5698306972863/1127160479744*x - 29043873093/953751175168
0
1
0:
0
0
1
1:
3/2
0
1
2:
5/2*x - 1/6
0
1
3:
35/8*x^2 - 5/12*x - 25/24
0
1
4:
63/8*x^3 - 7/8*x^2 - 91/24*x + 19/120
0
1
5:
231/16*x^4 - 7/4*x^3 - 21/2*x^2 + 7/10*x + 203/240
0
1
6:
429/16*x^5 - 55/16*x^4 - 209/8*x^3 + 87/40*x^2 + 381/80*x - 81/560
0
1
7:
6435/128*x^6 - 429/64*x^5 - 7865/128*x^4 + 187/32*x^3 + 11517/640*x^2 - 2067/2240*x - 3273/4480
0
1
8:
12155/128*x^7 - 5005/384*x^6 - 17875/128*x^5 + 1859/128*x^4 + 7293/128*x^3 - 3399/896*x^2 - 24959/4480*x + 5359/40320
0
1
9:
46189/256*x^8 - 2431/96*x^7 - 119119/384*x^6 + 11011/320*x^5 + 10439/64*x^4 - 715/56*x^3 - 23881/896*x^2 + 4477/4032*x + 52613/80640
0
1
10:
88179/256*x^9 - 12597/256*x^8 - 130169/192*x^7 + 75803/960*x^6 + 280007/640*x^5 - 4901/128*x^4 - 45851/448*x^3 + 7631/1344*x^2 + 101257/16128*x - 21877/177408
0
3/2
0:
0
0
3/2
1:
7/4
0
3/2
2:
99/32*x - 81/224
0
3/2
3:
715/128*x^2 - 429/448*x - 9691/8064
0
3/2
4:
20995/2048*x^3 - 29835/14336*x^2 - 587665/129024*x + 165919/473088
0
3/2
5:
156009/8192*x^4 - 8721/2048*x^3 - 159239/12288*x^2 + 253963/157696*x + 71686921/73801728
0
3/2
6:
2340135/65536*x^5 - 557175/65536*x^4 - 1077205/32768*x^3 + 5553985/1081344*x^2 + 52847645/9371648*x - 191140843/590413824
0
3/2
7:
17678835/262144*x^6 - 15397695/917504*x^5 - 143958725/1835008*x^4 + 70979725/5046272*x^3 + 5720148645/262406144*x^2 - 2508540569/1180827648*x - 11199859871/13382713344
0
3/2
8:
1074687075/8388608*x^7 - 276348105/8388608*x^6 - 10589001855/58720256*x^5 + 2082890925/58720256*x^4 + 53623578275/763363328*x^3 - 74893555075/8396996608*x^2 - 933694440603/142748942336*x + 7307055269771/24410069139456
0
3/2
9:
8205150525/33554432*x^8 - 1893496275/29360128*x^7 - 10172567075/25165824*x^6 + 356574865/4194304*x^5 + 3421526575/16777216*x^4 - 4991277725/163577856*x^3 - 59152427525/1853882368*x^2 + 495456669475/193730707456*x + 3469520177947/4649536978944
0
3/2
10:
125788525005/268435456*x^9 - 236950477335/1879048192*x^8 - 1254294010255/1409286144*x^7 + 13214890955/67108864*x^6 + 74264938115/134217728*x^5 - 37329046645/402653184*x^4 - 142007446935/1140850688*x^3 + 3750589928945/281790119936*x^2 + 24785478596185/3381481439232*x - 79530836198435/285171601375232
0
2
0:
0
0
2
1:
2
0
2
2:
15/4*x - 5/8
0
2
3:
7*x^2 - 7/4*x - 53/40
0
2
4:
105/8*x^3 - 63/16*x^2 - 21/4*x + 49/80
0
2
5:
99/4*x^4 - 33/4*x^3 - 1233/80*x^2 + 117/40*x + 589/560
0
2
6:
3003/64*x^5 - 2145/128*x^4 - 3201/80*x^3 + 3069/320*x^2 + 405/64*x - 2547/4480
0
2
7:
715/8*x^6 - 2145/64*x^5 - 12441/128*x^4 + 429/16*x^3 + 56331/2240*x^2 - 8613/2240*x - 12049/13440
0
2
8:
21879/128*x^7 - 17017/256*x^6 - 72501/320*x^5 + 17589/256*x^4 + 74217/896*x^3 - 29601/1792*x^2 - 231/32*x + 14179/26880
0
2
9:
20995/64*x^8 - 4199/32*x^7 - 131495/256*x^6 + 106743/640*x^5 + 438633/1792*x^4 - 25779/448*x^3 - 193973/5376*x^2 + 12415/2688*x + 234679/295680
0
2
10:
323323/512*x^9 - 264537/1024*x^8 - 365313/320*x^7 + 499681/1280*x^6 + 862563/1280*x^5 - 90831/512*x^4 - 3016/21*x^3 + 43927/1792*x^2 + 4095/512*x - 116519/236544
1/2
1
0:
0
1/2
1
1:
7/4
1/2
1
2:
99/32*x - 27/224
1/2
1
3:
715/128*x^2 - 143/448*x - 3553/2688
1/2
1
4:
20995/2048*x^3 - 9945/14336*x^2 - 211315/43008*x + 58045/473088
1/2
1
5:
156009/8192*x^4 - 2907/2048*x^3 - 56525/4096*x^2 + 88009/157696*x + 9070881/8200192
1/2
1
6:
2340135/65536*x^5 - 185725/65536*x^4 - 1136637/32768*x^3 + 1912483/1081344*x^2 + 177090887/28114944*x - 1083767/9371648
1/2
1
7:
17678835/262144*x^6 - 5132565/917504*x^5 - 150863325/1835008*x^4 + 24329975/5046272*x^3 + 6297932565/262406144*x^2 - 295621057/393609216*x - 12987088183/13382713344
1/2
1
8:
1074687075/8388608*x^7 - 92116035/8388608*x^6 - 11038106295/58720256*x^5 + 711505575/58720256*x^4 + 58386942675/763363328*x^3 - 26276367825/8396996608*x^2 - 1064045295459/142748942336*x + 293394898569/2712229904384
1/2
1
9:
8205150525/33554432*x^8 - 631165425/29360128*x^7 - 3519742325/8388608*x^6 + 121477375/4194304*x^5 + 25848834625/117440512*x^4 - 4062180075/381681664*x^3 - 465414655975/12977176576*x^2 + 1241151197075/1356114952192*x + 9491942480015/10848919617536
1/2
1
10:
125788525005/268435456*x^9 - 78983492445/1879048192*x^8 - 432494673325/469762048*x^7 + 4492334465/67108864*x^6 + 79577021235/134217728*x^5 - 4319888285/134217728*x^4 - 157842325055/1140850688*x^3 + 1332012797475/281790119936*x^2 + 561103724195/66303557632*x - 2632040630435/25924691034112
1/2
3/2
0:
0
1/2
3/2
1:
2
1/2
3/2
2:
15/4*x - 5/16
1/2
3/2
3:
7*x^2 - 7/8*x - 49/32
1/2
3/2
4:
105/8*x^3 - 63/32*x^2 - 189/32*x + 21/64
1/2
3/2
5:
99/4*x^4 - 33/8*x^3 - 1089/64*x^2 + 99/64*x + 165/128
1/2
3/2
6:
3003/64*x^5 - 2145/256*x^4 - 5577/128*x^3 + 1287/256*x^2 + 3861/512*x - 1287/4096
1/2
3/2
7:
715/8*x^6 - 2145/128*x^5 - 53625/512*x^4 + 3575/256*x^3 + 15015/512*x^2 - 2145/1024*x - 9295/8192
1/2
3/2
8:
21879/128*x^7 - 17017/512*x^6 - 123981/512*x^5 + 36465/1024*x^4 + 12155/128*x^3 - 36465/4096*x^2 - 36465/4096*x + 2431/8192
1/2
3/2
9:
20995/64*x^8 - 4199/64*x^7 - 558467/1024*x^6 + 88179/1024*x^5 + 566865/2048*x^4 - 62985/2048*x^3 - 356915/8192*x^2 + 20995/8192*x + 4199/4096
1/2
3/2
10:
323323/512*x^9 - 264537/2048*x^8 - 617253/512*x^7 + 205751/1024*x^6 + 3086265/4096*x^5 - 3086265/32768*x^4 - 2792335/16384*x^3 + 440895/32768*x^2 + 1322685/131072*x - 146965/524288
1/2
2
0:
0
1/2
2
1:
9/4
1/2
2
2:
143/32*x - 55/96
1/2
2
3:
1105/128*x^2 - 325/192*x - 2405/1408
1/2
2
4:
33915/2048*x^3 - 8075/2048*x^2 - 154445/22528*x + 179825/292864
1/2
2
5:
260015/8192*x^4 - 52003/6144*x^3 - 915705/45056*x^2 + 879529/292864*x + 5027155/3514368
1/2
2
6:
3991995/65536*x^5 - 1147125/65536*x^4 - 19164635/360448*x^3 + 140900599/14057472*x^2 + 242615275/28114944*x - 284138141/477954048
1/2
2
7:
30705345/262144*x^6 - 4652325/131072*x^5 - 34067025/262144*x^4 + 266227875/9371648*x^3 + 1288636065/37486592*x^2 - 1298610843/318636032*x - 15194769075/12108169216
1/2
2
8:
1893496275/8388608*x^7 - 597048375/8388608*x^6 - 2554126425/8388608*x^5 + 8025260625/109051904*x^4 + 12374834325/109051904*x^3 - 27762845175/1568669696*x^2 - 3912089003175/387461414912*x + 219042734795/387461414912
1/2
2
9:
14626572675/33554432*x^8 - 594576125/4194304*x^7 - 5815203975/8388608*x^6 + 755946625/4194304*x^5 + 5637157025/16777216*x^4 - 57653341975/926941184*x^3 - 1776824041225/35223764992*x^2 + 87943782825/17611882496*x + 3665854539475/3240586379264
1/2
2
10:
226419345009/268435456*x^9 - 75473115003/268435456*x^8 - 103768884687/67108864*x^7 + 28518598521/67108864*x^6 + 124379703903/134217728*x^5 - 440738618397/2281701376*x^4 - 4351535928537/21676163072*x^3 + 7522434606939/281790119936*x^2 + 294886455746841/25924691034112*x - 69545446101327/129623455170560
1
3/2
0:
0
1
3/2
1:
9/4
1
3/2
2:
143/32*x - 55/288
1
3/2
3:
1105/128*x^2 - 325/576*x - 23621/12672
1
3/2
4:
33915/2048*x^3 - 8075/6144*x^2 - 1496765/202752*x + 562475/2635776
1
3/2
5:
260015/8192*x^4 - 52003/18432*x^3 - 2927995/135168*x^2 + 910537/878592*x + 17116025/10543104
1
3/2
6:
3991995/65536*x^5 - 382375/65536*x^4 - 20265875/360448*x^3 + 48378085/14057472*x^2 + 813955345/84344832*x - 27476071/130351104
1
3/2
7:
30705345/262144*x^6 - 1550775/131072*x^5 - 35801225/262144*x^4 + 91065625/9371648*x^3 + 1423841105/37486592*x^2 - 316179781/220594176*x - 14427306665/9906683904
1
3/2
8:
1893496275/8388608*x^7 - 199016125/8388608*x^6 - 2670744705/8388608*x^5 + 2737117875/109051904*x^4 + 13538882725/109051904*x^3 - 126138453225/20392706048*x^2 - 4476822431535/387461414912*x + 235744411667/1162384244736
1
3/2
9:
14626572675/33554432*x^8 - 594576125/12582912*x^7 - 18167419025/25165824*x^6 + 257226375/4194304*x^5 + 18351489275/50331648*x^4 - 60150224575/2780823552*x^3 - 2009262570525/35223764992*x^2 + 79455518275/44707086336*x + 142423497632675/106939350515712
1
3/2
10:
226419345009/268435456*x^9 - 25157705001/268435456*x^8 - 323078828917/201326592*x^7 + 29056926913/201326592*x^6 + 134047262223/134217728*x^5 - 457985365229/6845104128*x^4 - 14613553695923/65028489216*x^3 + 2660324146977/281790119936*x^2 + 1028188006465067/77774073102336*x - 75427567007023/388870365511680
1
2
0:
0
1
2
1:
5/2
1
2
2:
21/4*x - 9/20
1
2
3:
21/2*x^2 - 7/5*x - 21/10
1
2
4:
165/8*x^3 - 27/8*x^2 - 69/8*x + 29/56
1
2
5:
1287/32*x^4 - 297/40*x^3 - 2079/80*x^2 + 729/280*x + 2061/1120
1
2
6:
5005/64*x^5 - 1001/64*x^4 - 11011/160*x^3 + 1419/160*x^2 + 3597/320*x - 499/960
1
2
7:
2431/16*x^6 - 1287/40*x^5 - 2717/16*x^4 + 715/28*x^3 + 25311/560*x^2 - 3047/840*x - 2783/1680
1
2
8:
37791/128*x^7 - 41769/640*x^6 - 256581/640*x^5 + 60021/896*x^4 + 134667/896*x^3 - 14313/896*x^2 - 60177/4480*x + 24897/49280
1
2
9:
146965/256*x^8 - 4199/32*x^7 - 29393/32*x^6 + 663/4*x^5 + 401115/896*x^4 - 38155/672*x^3 - 22685/336*x^2 + 16705/3696*x + 89791/59136
1
2
10:
572033/512*x^9 - 671517/2560*x^8 - 1318163/640*x^7 + 252263/640*x^6 + 1588191/1280*x^5 - 136697/768*x^4 - 725747/2688*x^3 + 21877/896*x^2 + 165029/10752*x - 68009/139776
3/2
2
0:
0
3/2
2
1:
11/4
3/2
2
2:
195/32*x - 91/352
3/2
2
3:
1615/128*x^2 - 595/704*x - 45535/18304
3/2
2
4:
52003/2048*x^3 - 47481/22528*x^2 - 3078513/292864*x + 92225/292864
3/2
2
5:
412965/8192*x^4 - 107065/22528*x^3 - 18932151/585728*x^2 + 477603/292864*x + 45017707/19914752
3/2
2
6:
6513255/65536*x^5 - 7353675/720896*x^4 - 407277675/4685824*x^3 + 26614105/4685824*x^2 + 2240612045/159318016*x - 986468441/3027042304
3/2
2
7:
51175575/262144*x^6 - 2791395/131072*x^5 - 739719675/3407872*x^4 + 155927925/9371648*x^3 + 36543437415/637272064*x^2 - 14053963815/6054084608*x - 25176506997/12108169216
3/2
2
8:
3210711075/8388608*x^7 - 366724575/8388608*x^6 - 4340619225/8388608*x^5 + 4826079225/109051904*x^4 + 357416486325/1853882368*x^3 - 4025892825225/387461414912*x^2 - 6619451034375/387461414912*x + 2877166580325/8911612542976
3/2
2
9:
25157705001/33554432*x^8 - 372312759/4194304*x^7 - 10024503573/8388608*x^6 + 464517795/4194304*x^5 + 165251002035/285212672*x^4 - 659058357279/17611882496*x^3 - 179788198827/2071986176*x^2 + 1188274677549/405073297408*x + 31325132567037/16202931896320
3/2
2
10:
394137378349/268435456*x^9 - 48028345911/268435456*x^8 - 181029919203/67108864*x^7 + 17842372989/67108864*x^6 + 3691292404275/2281701376*x^5 - 5141493210483/43352326144*x^4 - 7585832041989/21676163072*x^3 + 104021716918113/6481172758528*x^2 + 2559060216172233/129623455170560*x - 122505679556113/388870365511680
Definition
For $n\geq0$ and $\alpha,\beta>-1$, $q_n^{(\alpha,\beta)}$ is the secondary polynomial of the Jacobi polynomial $P_n^{(\alpha,\beta)}$, taken with respect to the probability-normalised Jacobi weight on $[-1,1]$ [1]. The table uses $q_0^{(\alpha,\beta)}=0$.
Parameters
$\alpha$
—   real number ($\alpha>-1$)
$\beta$
—   real number ($\beta>-1$)
$n$
—   integer ($n\geq0$)
Formulas
(1)
$q_n^{(\alpha,\beta)}(x)=\int_{-1}^{1}\dfrac{P_n^{(\alpha,\beta)}(t)-P_n^{(\alpha,\beta)}(x)}{t-x}\rho_{\alpha,\beta}(t)\,\mathrm{d}t$, where $\rho_{\alpha,\beta}(t)=\dfrac{(1-t)^\alpha(1+t)^\beta}{2^{\alpha+\beta+1}B(\alpha+1,\beta+1)}$.
(2)
If $P_n^{(\alpha,\beta)}(x)=\sum_{j=0}^n c_jx^j$, then $q_n^{(\alpha,\beta)}(x)=\sum_{j=1}^n c_j\sum_{i=0}^{j-1}m_i x^{j-1-i}$, where $m_i=\sum_{k=0}^{i}\binom{i}{k}2^k(-1)^{i-k}\dfrac{(\beta+1)_k}{(\alpha+\beta+2)_k}$ is the $i$-th moment of the probability density $\rho_{\alpha,\beta}$.
(3)
$q_0^{(\alpha,\beta)}=0$, $q_1^{(\alpha,\beta)}=(\alpha+\beta+2)/2$, and for $n\geq2$ the $q_n^{(\alpha,\beta)}$ satisfy the same three-term recurrence as the Jacobi polynomials: $2n(n+\alpha+\beta)(2n+\alpha+\beta-2)q_n=(2n+\alpha+\beta-1)((2n+\alpha+\beta)(2n+\alpha+\beta-2)x+\alpha^2-\beta^2)q_{n-1}-2(n+\alpha-1)(n+\beta-1)(2n+\alpha+\beta)q_{n-2}$.
(4)
If $x_k$ is a root of $P_n^{(\alpha,\beta)}$, then $w_k=\dfrac{q_n^{(\alpha,\beta)}(x_k)}{(P_n^{(\alpha,\beta)})'(x_k)}$ and $\sum_{k=1}^n w_k=1$ for the probability-normalised Gauss-Jacobi rule.
(5)
$q_n^{(\alpha,\beta)}(z)/P_n^{(\alpha,\beta)}(z)$ is the $[n-1/n]$ Pade approximant at $z=\infty$ to $\int_{-1}^{1}(z-t)^{-1}\rho_{\alpha,\beta}(t)\,\mathrm{d}t$, the Stieltjes transform of the probability density [2].
Comments
(6)
$P_n^{(\alpha,\beta)}$ is the Jacobi polynomial normalised by $P_n^{(\alpha,\beta)}(1)=\binom{n+\alpha}{n}$, as in the Jacobi-polynomial table. It is orthogonal for the weight $(1-t)^\alpha(1+t)^\beta$ on $[-1,1]$.
(7)
The normalising constant of the Jacobi weight is $2^{\alpha+\beta+1}B(\alpha+1,\beta+1)$. Dividing by it makes $q_1^{(\alpha,\beta)}=(\alpha+\beta+2)/2$ and keeps every stored entry in $\mathbb{Q}[x]$. The unnormalised weight would multiply each entry by the normalising constant, which is not rational for most rational $\alpha,\beta$.
(8)
The stored grid follows the rational entries of the Jacobi-polynomial table: $\alpha,\beta\in\{-\frac12,0,\frac12,1,\frac32,2\}$ with $\alpha<\beta$, and $0\leq n\leq10$. The symbolic $a,b$ rows of that table are not repeated, because after probability normalisation the coefficients of $q_n^{(a,b)}$ are rational functions of $a$ and $b$, not polynomials over $\mathbb{Q}$.
(9)
The Legendre case $\alpha=\beta=0$ is not stored here. The Legendre secondary-polynomial table uses the density $1$ on $[-1,1]$, whose total mass is $2$, so it holds twice the probability-normalised values.
(10)
At the roots $x_1<\cdots<x_n$ of $P_n^{(\alpha,\beta)}$, $q_n^{(\alpha,\beta)}(x_k)/(P_n^{(\alpha,\beta)})'(x_k)$ is the weight of the probability-normalised $n$-point Gauss-Jacobi quadrature rule. A general Gauss-Jacobi quadrature table is not in this corpus.
(11)
These are also the numerator polynomials, or associated polynomials of order one, of the Jacobi family. They are not the associated Jacobi polynomials obtained by changing $\alpha$ and $\beta$.
Links
Similar tables
Jacobi polynomials —   the $P_n^{(\alpha,\beta)}$ these are the secondary polynomials of, and the recurrence they share
Secondary polynomials of the Legendre polynomials —   the Legendre case, stored with density $1$ rather than probability density $\frac12$
Secondary polynomials of the Laguerre polynomials —   the same probability-normalised construction for the Laguerre weight on $[0,\infty)$
Data properties
Entries are of type: rational polynomial
How well the digits are known: exact
Table is complete: no (it holds the rational Jacobi-polynomial grid of T106 with $\alpha<\beta$, $\alpha,\beta\in\{-\frac12,0,\frac12,1,\frac32,2\}$ and $n\leq10$)
How they were obtained:

Every value is exact, a polynomial with rational coefficients, so there is no precision to choose. The generator computes $P_n^{(\alpha,\beta)}$ from its binomial formula and evaluates the defining integral from the exact moments of the probability density.

more

Before the entries are written, they are checked against the same three-term recurrence as the Jacobi polynomials, against an independent plain Python fractions computation, against direct rational quadrature for a control case, and against the probability-normalised Legendre relation to T133.



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