Legendre polynomials
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Polynomials
$n$ 
$P_n(x)$
0:
1
1:
x
2:
3/2*x^2 - 1/2
3:
5/2*x^3 - 3/2*x
4:
35/8*x^4 - 15/4*x^2 + 3/8
5:
63/8*x^5 - 35/4*x^3 + 15/8*x
6:
231/16*x^6 - 315/16*x^4 + 105/16*x^2 - 5/16
7:
429/16*x^7 - 693/16*x^5 + 315/16*x^3 - 35/16*x
8:
6435/128*x^8 - 3003/32*x^6 + 3465/64*x^4 - 315/32*x^2 + 35/128
9:
12155/128*x^9 - 6435/32*x^7 + 9009/64*x^5 - 1155/32*x^3 + 315/128*x
10:
46189/256*x^10 - 109395/256*x^8 + 45045/128*x^6 - 15015/128*x^4 + 3465/256*x^2 - 63/256
11:
88179/256*x^11 - 230945/256*x^9 + 109395/128*x^7 - 45045/128*x^5 + 15015/256*x^3 - 693/256*x
12:
676039/1024*x^12 - 969969/512*x^10 + 2078505/1024*x^8 - 255255/256*x^6 + 225225/1024*x^4 - 9009/512*x^2 + 231/1024
13:
1300075/1024*x^13 - 2028117/512*x^11 + 4849845/1024*x^9 - 692835/256*x^7 + 765765/1024*x^5 - 45045/512*x^3 + 3003/1024*x
14:
5014575/2048*x^14 - 16900975/2048*x^12 + 22309287/2048*x^10 - 14549535/2048*x^8 + 4849845/2048*x^6 - 765765/2048*x^4 + 45045/2048*x^2 - 429/2048
15:
9694845/2048*x^15 - 35102025/2048*x^13 + 50702925/2048*x^11 - 37182145/2048*x^9 + 14549535/2048*x^7 - 2909907/2048*x^5 + 255255/2048*x^3 - 6435/2048*x
16:
300540195/32768*x^16 - 145422675/4096*x^14 + 456326325/8192*x^12 - 185910725/4096*x^10 + 334639305/16384*x^8 - 20369349/4096*x^6 + 4849845/8192*x^4 - 109395/4096*x^2 + 6435/32768
17:
583401555/32768*x^17 - 300540195/4096*x^15 + 1017958725/8192*x^13 - 456326325/4096*x^11 + 929553625/16384*x^9 - 66927861/4096*x^7 + 20369349/8192*x^5 - 692835/4096*x^3 + 109395/32768*x
18:
2268783825/65536*x^18 - 9917826435/65536*x^16 + 4508102925/16384*x^14 - 4411154475/16384*x^12 + 5019589575/32768*x^10 - 1673196525/32768*x^8 + 156165009/16384*x^6 - 14549535/16384*x^4 + 2078505/65536*x^2 - 12155/65536
19:
4418157975/65536*x^19 - 20419054425/65536*x^17 + 9917826435/16384*x^15 - 10518906825/16384*x^13 + 13233463425/32768*x^11 - 5019589575/32768*x^9 + 557732175/16384*x^7 - 66927861/16384*x^5 + 14549535/65536*x^3 - 230945/65536*x
20:
34461632205/262144*x^20 - 83945001525/131072*x^18 + 347123925225/262144*x^16 - 49589132175/32768*x^14 + 136745788725/131072*x^12 - 29113619535/65536*x^10 + 15058768725/131072*x^8 - 557732175/32768*x^6 + 334639305/262144*x^4 - 4849845/131072*x^2 + 46189/262144
21:
67282234305/262144*x^21 - 172308161025/131072*x^19 + 755505013725/262144*x^17 - 115707975075/32768*x^15 + 347123925225/131072*x^13 - 82047473235/65536*x^11 + 48522699225/131072*x^9 - 2151252675/32768*x^7 + 1673196525/262144*x^5 - 37182145/131072*x^3 + 969969/262144*x
22:
263012370465/524288*x^22 - 1412926920405/524288*x^20 + 3273855059475/524288*x^18 - 4281195077775/524288*x^16 + 1735619626125/262144*x^14 - 902522205585/262144*x^12 + 300840735195/262144*x^10 - 62386327575/262144*x^8 + 15058768725/524288*x^6 - 929553625/524288*x^4 + 22309287/524288*x^2 - 88179/524288
23:
514589420475/524288*x^23 - 2893136075115/524288*x^21 + 7064634602025/524288*x^19 - 9821565178425/524288*x^17 + 4281195077775/262144*x^15 - 2429867476575/262144*x^13 + 902522205585/262144*x^11 - 214886239425/262144*x^9 + 62386327575/524288*x^7 - 5019589575/524288*x^5 + 185910725/524288*x^3 - 2028117/524288*x
24:
8061900920775/4194304*x^24 - 11835556670925/1048576*x^22 + 60755857577415/2097152*x^20 - 44742685812825/1048576*x^18 + 166966608033225/4194304*x^16 - 12843585233325/524288*x^14 + 10529425731825/1048576*x^12 - 1418249180205/524288*x^10 + 1933976154825/4194304*x^8 - 48522699225/1048576*x^6 + 5019589575/2097152*x^4 - 50702925/1048576*x^2 + 676039/4194304
25:
15801325804719/4194304*x^25 - 24185702762325/1048576*x^23 + 130191123380175/2097152*x^21 - 101259762629025/1048576*x^19 + 402684172315425/4194304*x^17 - 33393321606645/524288*x^15 + 29968365544425/1048576*x^13 - 4512611027925/524288*x^11 + 7091245901025/4194304*x^9 - 214886239425/1048576*x^7 + 29113619535/2097152*x^5 - 456326325/1048576*x^3 + 16900975/4194304*x
26:
61989816618513/8388608*x^26 - 395033145117975/8388608*x^24 + 556271163533475/4194304*x^22 - 911337863661225/4194304*x^20 + 1923935489951475/8388608*x^18 - 1369126185872445/8388608*x^16 + 166966608033225/2097152*x^14 - 55655536011075/2097152*x^12 + 49638721307175/8388608*x^10 - 7091245901025/8388608*x^8 + 300840735195/4194304*x^6 - 13233463425/4194304*x^4 + 456326325/8388608*x^2 - 1300075/8388608
27:
121683714103007/8388608*x^27 - 805867616040669/8388608*x^25 + 1185099435353925/4194304*x^23 - 2039660932956075/4194304*x^21 + 4556689318306125/8388608*x^19 - 3463083881912655/8388608*x^17 + 456375395290815/2097152*x^15 - 166966608033225/2097152*x^13 + 166966608033225/8388608*x^11 - 27577067392875/8388608*x^9 + 1418249180205/4194304*x^7 - 82047473235/4194304*x^5 + 4411154475/8388608*x^3 - 35102025/8388608*x
28:
956086325095055/33554432*x^28 - 3285460280781189/16777216*x^26 + 20146690401016725/33554432*x^24 - 9085762337713425/8388608*x^22 + 42832879592077575/33554432*x^20 - 17315419409563275/16777216*x^18 + 19624141997505045/33554432*x^16 - 977947275623175/4194304*x^14 + 2170565904431925/33554432*x^12 - 204070298707275/16777216*x^10 + 49638721307175/33554432*x^8 - 902522205585/8388608*x^6 + 136745788725/33554432*x^4 - 1017958725/16777216*x^2 + 5014575/33554432
29:
1879204156221315/33554432*x^29 - 6692604275665385/16777216*x^27 + 42710983650155457/33554432*x^25 - 20146690401016725/8388608*x^23 + 99943385714847675/33554432*x^21 - 42832879592077575/16777216*x^19 + 51946258228689825/33554432*x^17 - 2803448856786435/4194304*x^15 + 6845630929362225/33554432*x^13 - 723521968143975/16777216*x^11 + 204070298707275/33554432*x^9 - 4512611027925/8388608*x^7 + 902522205585/33554432*x^5 - 10518906825/16777216*x^3 + 145422675/33554432*x
30:
7391536347803839/67108864*x^30 - 54496920530418135/67108864*x^28 + 180700315442965395/67108864*x^26 - 355924863751295475/67108864*x^24 + 463373879223384675/67108864*x^22 - 419762220002360235/67108864*x^20 + 271274904083157975/67108864*x^18 - 126155198555389575/67108864*x^16 + 42051732851796525/67108864*x^14 - 9888133564634325/67108864*x^12 + 1591748329916745/67108864*x^10 - 166966608033225/67108864*x^8 + 10529425731825/67108864*x^6 - 347123925225/67108864*x^4 + 4508102925/67108864*x^2 - 9694845/67108864
31:
14544636039226909/67108864*x^31 - 110873045217057585/67108864*x^29 + 381478443712926945/67108864*x^27 - 783034700252850045/67108864*x^25 + 1067774591253886425/67108864*x^23 - 1019422534291446285/67108864*x^21 + 699603700003933725/67108864*x^19 - 348782019535488825/67108864*x^17 + 126155198555389575/67108864*x^15 - 32706903329175075/67108864*x^13 + 5932880138780595/67108864*x^11 - 723521968143975/67108864*x^9 + 55655536011075/67108864*x^7 - 2429867476575/67108864*x^5 + 49589132175/67108864*x^3 - 300540195/67108864*x
32:
916312070471295267/2147483648*x^32 - 450883717216034179/134217728*x^30 + 3215318311294669965/268435456*x^28 - 3433305993416342505/134217728*x^26 + 19575867506321251125/536870912*x^24 - 4911763119767877555/134217728*x^22 + 7135957740040123995/268435456*x^20 - 1898924328582105825/134217728*x^18 + 5929294332103310025/1073741824*x^16 - 210258664258982625/134217728*x^14 + 85037948655855195/268435456*x^12 - 5932880138780595/134217728*x^10 + 2170565904431925/536870912*x^8 - 29968365544425/134217728*x^6 + 1735619626125/268435456*x^4 - 9917826435/134217728*x^2 + 300540195/2147483648
33:
1804857108504066435/2147483648*x^33 - 916312070471295267/134217728*x^31 + 6763255758240512685/268435456*x^29 - 7502409393020896585/134217728*x^27 + 44632977914412452565/536870912*x^25 - 11745520503792750675/134217728*x^23 + 18009798105815551035/268435456*x^21 - 5097112671457231425/134217728*x^19 + 17090318957238952425/1073741824*x^17 - 658810481344812225/134217728*x^15 + 294362129962575675/268435456*x^13 - 23192167815233235/134217728*x^11 + 9888133564634325/536870912*x^9 - 166966608033225/134217728*x^7 + 12843585233325/268435456*x^5 - 115707975075/134217728*x^3 + 9917826435/2147483648*x
34:
7113260368810144185/4294967296*x^34 - 59560284580634192355/4294967296*x^32 + 28405674184610153277/536870912*x^30 - 65378138996324955955/536870912*x^28 + 202565053611564207795/1073741824*x^26 - 223164889572062262825/1073741824*x^24 + 90048990529077755175/536870912*x^22 - 54029394317446653105/536870912*x^20 + 96845140757687397075/2147483648*x^18 - 32281713585895799025/2147483648*x^16 + 1976431444034436675/536870912*x^14 - 347882517228498525/536870912*x^12 + 85037948655855195/1073741824*x^10 - 6845630929362225/1073741824*x^8 + 166966608033225/536870912*x^6 - 4281195077775/536870912*x^4 + 347123925225/4294967296*x^2 - 583401555/4294967296
35:
14023284727082855679/4294967296*x^35 - 120925426269772451145/4294967296*x^33 + 59560284580634192355/536870912*x^31 - 142028370923050766385/536870912*x^29 + 457646972974274691685/1073741824*x^27 - 526669139390066940267/1073741824*x^25 + 223164889572062262825/536870912*x^23 - 141505556545693615275/536870912*x^21 + 270146971587233265525/2147483648*x^19 - 96845140757687397075/2147483648*x^17 + 6456342717179159805/536870912*x^15 - 1257729100749186975/536870912*x^13 + 347882517228498525/1073741824*x^11 - 32706903329175075/1073741824*x^9 + 977947275623175/536870912*x^7 - 33393321606645/536870912*x^5 + 4281195077775/4294967296*x^3 - 20419054425/4294967296*x
36:
110628135069209194801/17179869184*x^36 - 490814965447899948765/8589934592*x^34 + 3990539066902490887785/17179869184*x^32 - 615456273999886654335/1073741824*x^30 + 4118822756768472225165/4294967296*x^28 - 2471293654061083335099/2147483648*x^26 + 4388909494917224502225/4294967296*x^24 - 733256065736776006425/1073741824*x^22 + 2971616687459565920775/8589934592*x^20 - 570310273350825782775/4294967296*x^18 + 329273478576137150055/8589934592*x^16 - 8804103705244308825/1073741824*x^14 + 5450159436579810225/4294967296*x^12 - 294362129962575675/2147483648*x^10 + 42051732851796525/4294967296*x^8 - 456375395290815/1073741824*x^6 + 166966608033225/17179869184*x^4 - 755505013725/8589934592*x^2 + 2268783825/17179869184
37:
218266320541953276229/17179869184*x^37 - 995653215622882753209/8589934592*x^35 + 8343854412614299129005/17179869184*x^33 - 1330179688967496962595/1073741824*x^31 + 9231844109998299815025/4294967296*x^29 - 5766351859475861115231/2147483648*x^27 + 10708939167598027785429/4294967296*x^25 - 1880961212107381929525/1073741824*x^23 + 8065816723104536070675/8589934592*x^21 - 1650898159699758844875/4294967296*x^19 + 1026558492031486408995/8589934592*x^17 - 29933952597830650005/1073741824*x^15 + 20542908645570053925/4294967296*x^13 - 1257729100749186975/2147483648*x^11 + 210258664258982625/4294967296*x^9 - 2803448856786435/1073741824*x^7 + 1369126185872445/17179869184*x^5 - 9821565178425/8589934592*x^3 + 83945001525/17179869184*x
38:
861577581086657669325/34359738368*x^38 - 8075853860052271220473/34359738368*x^36 + 34847862546800896362315/34359738368*x^34 - 91782398538757290419055/34359738368*x^32 + 41235570357992405840445/8589934592*x^30 - 53544695837990138927145/8589934592*x^28 + 51897166735282750037079/8589934592*x^26 - 38246211312850099233675/8589934592*x^24 + 43262107878469784379075/17179869184*x^22 - 18820239020577250831575/17179869184*x^20 + 6273413006859083610525/17179869184*x^18 - 1586499487685024450265/17179869184*x^16 + 149669762989153250025/8589934592*x^14 - 20542908645570053925/8589934592*x^12 + 1976431444034436675/8589934592*x^10 - 126155198555389575/8589934592*x^8 + 19624141997505045/34359738368*x^6 - 402684172315425/34359738368*x^4 + 3273855059475/34359738368*x^2 - 4418157975/34359738368
39:
1701063429324939500975/34359738368*x^39 - 16369974040646495717175/34359738368*x^37 + 72682684740470440984257/34359738368*x^35 - 197471221098538412719785/34359738368*x^33 + 91782398538757290419055/8589934592*x^31 - 123706711073977217521335/8589934592*x^29 + 124937623621976990830005/8589934592*x^27 - 96380452508382250068861/8589934592*x^25 + 114738633938550297701025/17179869184*x^23 - 52875909629240847574425/17179869184*x^21 + 18820239020577250831575/17179869184*x^19 - 5132792460157432044975/17179869184*x^17 + 528833162561674816755/8589934592*x^15 - 80591410840313288475/8589934592*x^13 + 8804103705244308825/8589934592*x^11 - 658810481344812225/8589934592*x^9 + 126155198555389575/34359738368*x^7 - 3463083881912655/34359738368*x^5 + 44742685812825/34359738368*x^3 - 172308161025/34359738368*x
40:
26876802183334044115405/274877906944*x^40 - 66341473743672640538025/68719476736*x^38 + 605689039503920341535475/137438953472*x^36 - 847964655305488478149665/68719476736*x^34 + 6516550296251767619752905/274877906944*x^32 - 569050870940295200598141/17179869184*x^30 + 1195831540381779769372905/34359738368*x^28 - 481902262541911250344305/17179869184*x^26 + 2409511312709556251721525/137438953472*x^24 - 293220953398517427458175/34359738368*x^22 + 222078820442811559812585/68719476736*x^20 - 32507685580997069618175/34359738368*x^18 + 29085823940892114921525/137438953472*x^16 - 610192110648086327025/17179869184*x^14 + 149669762989153250025/34359738368*x^12 - 6456342717179159805/17179869184*x^10 + 5929294332103310025/274877906944*x^8 - 51946258228689825/68719476736*x^6 + 1923935489951475/137438953472*x^4 - 7064634602025/68719476736*x^2 + 34461632205/274877906944
41:
53098072606098965203605/274877906944*x^41 - 134384010916670220577025/68719476736*x^39 + 1260488001129780170222475/137438953472*x^37 - 1817067118511761024606425/68719476736*x^35 + 14415399140193304128544305/274877906944*x^33 - 1303310059250353523950581/17179869184*x^31 + 2845254354701476002990705/34359738368*x^29 - 1195831540381779769372905/17179869184*x^27 + 6264729413044846254475965/137438953472*x^25 - 803170437569852083907175/34359738368*x^23 + 645086097476738340407985/68719476736*x^21 - 100944918383096163551175/34359738368*x^19 + 97523056742991208854525/137438953472*x^17 - 2237371072376316532425/17179869184*x^15 + 610192110648086327025/34359738368*x^13 - 29933952597830650005/17179869184*x^11 + 32281713585895799025/274877906944*x^9 - 348782019535488825/68719476736*x^7 + 17315419409563275/137438953472*x^5 - 101259762629025/68719476736*x^3 + 1412926920405/274877906944*x
42:
209863810776486386280915/549755813888*x^42 - 2177020976850057573347805/549755813888*x^40 + 5240976425750138602503975/274877906944*x^38 - 15546018680600622099410525/274877906944*x^36 + 63597349147911635861224875/549755813888*x^34 - 95141634325275807248392413/549755813888*x^32 + 13467537278920319747489337/68719476736*x^30 - 11787482326620400583818635/68719476736*x^28 + 32287451590308053773068435/274877906944*x^26 - 17402026147346795151322125/274877906944*x^24 + 3694584012821319585973005/137438953472*x^22 - 1231528004273773195324335/137438953472*x^20 + 639317816426275702490775/274877906944*x^18 - 127530151125450042348225/274877906944*x^16 + 4794366583663535426625/68719476736*x^14 - 528833162561674816755/68719476736*x^12 + 329273478576137150055/549755813888*x^10 - 17090318957238952425/549755813888*x^8 + 271274904083157975/274877906944*x^6 - 4556689318306125/274877906944*x^4 + 60755857577415/549755813888*x^2 - 67282234305/549755813888
43:
414847067813984717066925/549755813888*x^43 - 4407140026306214111899215/549755813888*x^41 + 10885104884250287866739025/274877906944*x^39 - 33192850696417544482525175/274877906944*x^37 + 139914168125405598894694725/549755813888*x^35 - 216230987102899561928164575/549755813888*x^33 + 31713878108425269082797471/68719476736*x^31 - 28859008454829256601762865/68719476736*x^29 + 82512376286342804086730445/274877906944*x^27 - 46637430074889411005543295/274877906944*x^25 + 10441215688408077090793275/137438953472*x^23 - 3694584012821319585973005/137438953472*x^21 + 2052546673789621992207225/274877906944*x^19 - 442604642141267794032075/274877906944*x^17 + 18218593017921434621175/68719476736*x^15 - 2237371072376316532425/68719476736*x^13 + 1586499487685024450265/549755813888*x^11 - 96845140757687397075/549755813888*x^9 + 1898924328582105825/274877906944*x^7 - 42832879592077575/274877906944*x^5 + 911337863661225/549755813888*x^3 - 2893136075115/549755813888*x
44:
3281063172710606398620225/2199023255552*x^44 - 17838423916001342833877775/1099511627776*x^42 + 180692741078554778587867815/2199023255552*x^40 - 141506363495253742267607325/549755813888*x^38 + 1228135475767449145853431475/2199023255552*x^36 - 979399176877839192262863075/1099511627776*x^34 + 2378540858131895181209810325/2199023255552*x^32 - 140447174480169048795245943/137438953472*x^30 + 836911245190048441451123085/1099511627776*x^28 - 247537128859028412260191335/549755813888*x^26 + 233187150374447055027716475/1099511627776*x^24 - 21831632803035070280749575/274877906944*x^22 + 25862088089749237101811035/1099511627776*x^20 - 2999875907846370603995175/549755813888*x^18 + 1074896988057364642649325/1099511627776*x^16 - 18218593017921434621175/137438953472*x^14 + 29085823940892114921525/2199023255552*x^12 - 1026558492031486408995/1099511627776*x^10 + 96845140757687397075/2199023255552*x^8 - 699603700003933725/549755813888*x^6 + 42832879592077575/2199023255552*x^4 - 130191123380175/1099511627776*x^2 + 263012370465/2199023255552
45:
6489213830472088210604445/2199023255552*x^45 - 36091694899816670384822475/1099511627776*x^43 + 374606902236028199511433275/2199023255552*x^41 - 301154568464257964313113025/549755813888*x^39 + 2688620906409821103084539175/2199023255552*x^37 - 2210643856381408462536176655/1099511627776*x^35 + 5549928668974422089489557425/2199023255552*x^33 - 339791551161699311601401475/137438953472*x^31 + 2106707617202535731928689145/1099511627776*x^29 - 650930968481148787795317955/549755813888*x^27 + 643596535033473871876497471/1099511627776*x^25 - 63596495556667378643922675/274877906944*x^23 + 80049320277795257696081775/1099511627776*x^21 - 9946956957595860423773475/549755813888*x^19 + 3856983310088190776565225/1099511627776*x^17 - 71659799203824309509955/137438953472*x^15 + 127530151125450042348225/2199023255552*x^13 - 5132792460157432044975/1099511627776*x^11 + 570310273350825782775/2199023255552*x^9 - 5097112671457231425/549755813888*x^7 + 419762220002360235/2199023255552*x^5 - 2039660932956075/1099511627776*x^3 + 11835556670925/2199023255552*x
46:
25674715590128696833261065/4398046511104*x^46 - 292014622371243969477200025/4398046511104*x^44 + 1551942880692116826547366425/4398046511104*x^42 - 5119627663892385393322921425/4398046511104*x^40 + 11745028170106060608211407975/4398046511104*x^38 - 19895794707432676162825589895/4398046511104*x^36 + 25790844991116432062922060975/4398046511104*x^34 - 26163949439450846993307913575/4398046511104*x^32 + 10533538086012678659643445725/2199023255552*x^30 - 6788280099874837358436887245/2199023255552*x^28 + 3515027229798203454094716957/2199023255552*x^26 - 1462719397803349708810221525/2199023255552*x^24 + 487573132601116569603407175/2199023255552*x^22 - 129310440448746185509055175/2199023255552*x^20 + 26998883170617335435956575/2199023255552*x^18 - 4371247751433282880107255/2199023255552*x^16 + 1074896988057364642649325/4398046511104*x^14 - 97523056742991208854525/4398046511104*x^12 + 6273413006859083610525/4398046511104*x^10 - 270146971587233265525/4398046511104*x^8 + 7135957740040123995/4398046511104*x^6 - 99943385714847675/4398046511104*x^4 + 556271163533475/4398046511104*x^2 - 514589420475/4398046511104
47:
50803160635786570329644235/4398046511104*x^47 - 590518458572960027165004495/4398046511104*x^45 + 3212160846083683664249200275/4398046511104*x^43 - 10863600164844817785831564975/4398046511104*x^41 + 25598138319461926966614607125/4398046511104*x^39 - 44631107046403030311203350305/4398046511104*x^37 + 59687384122298028488476769685/4398046511104*x^35 - 62634909264139906438525005225/4398046511104*x^33 + 26163949439450846993307913575/2199023255552*x^31 - 17555896810021131099405742875/2199023255552*x^29 + 9503592139824772301811642143/2199023255552*x^27 - 4154123089761513173021029131/2199023255552*x^25 + 1462719397803349708810221525/2199023255552*x^23 - 412561881431714020433652225/2199023255552*x^21 + 92364600320532989649325125/2199023255552*x^19 - 16199329902370401261573945/2199023255552*x^17 + 4371247751433282880107255/4398046511104*x^15 - 442604642141267794032075/4398046511104*x^13 + 32507685580997069618175/4398046511104*x^11 - 1650898159699758844875/4398046511104*x^9 + 54029394317446653105/4398046511104*x^7 - 1019422534291446285/4398046511104*x^5 + 9085762337713425/4398046511104*x^3 - 24185702762325/4398046511104*x
48:
1608766753466574727105400775/70368744177664*x^48 - 2387748549881968805493279045/8796093022208*x^46 + 26573330635783201222425202275/17592186044416*x^44 - 46040972127199465854238537275/8796093022208*x^42 + 445407606758637529219094163975/35184372088832*x^40 - 199665478891803030339593935575/8796093022208*x^38 + 550450320238970707171507987095/17592186044416*x^36 - 298436920611490142442383848425/8796093022208*x^34 + 2066952005716616912471325172425/70368744177664*x^32 - 90120270291441806310282813425/4398046511104*x^30 + 101824201498122560376553308675/8796093022208*x^28 - 23326998888660804740810394351/4398046511104*x^26 + 34617692414679276441841909425/17592186044416*x^24 - 2587888165344387946356545775/4398046511104*x^22 + 1237685644295142061300956675/8796093022208*x^20 - 116995160406008453555811825/4398046511104*x^18 + 275388608340296821446757065/70368744177664*x^16 - 3856983310088190776565225/8796093022208*x^14 + 639317816426275702490775/17592186044416*x^12 - 18820239020577250831575/8796093022208*x^10 + 2971616687459565920775/35184372088832*x^8 - 18009798105815551035/8796093022208*x^6 + 463373879223384675/17592186044416*x^4 - 1185099435353925/8796093022208*x^2 + 8061900920775/70368744177664
49:
3184701532372607112841303575/70368744177664*x^49 - 4826300260399724181316202325/8796093022208*x^47 + 54918216647285282526345418035/17592186044416*x^45 - 97435545664538404482225741675/8796093022208*x^43 + 966860414671188782939009282775/35184372088832*x^41 - 445407606758637529219094163975/8796093022208*x^39 + 1264548032981419192150761591975/17592186044416*x^37 - 707721840307248052077653126265/8796093022208*x^35 + 5073427650395332421520525423225/70368744177664*x^33 - 229661333968512990274591685825/4398046511104*x^31 + 270360810874325418930848440275/8796093022208*x^29 - 64797219135168902057806650975/4398046511104*x^27 + 101083661850863487210178375521/17592186044416*x^25 - 7988698249541371486578902175/4398046511104*x^23 + 4066681402684038201417429075/8796093022208*x^21 - 412561881431714020433652225/4398046511104*x^19 + 1052956443654076082002306425/70368744177664*x^17 - 16199329902370401261573945/8796093022208*x^15 + 2999875907846370603995175/17592186044416*x^13 - 100944918383096163551175/8796093022208*x^11 + 18820239020577250831575/35184372088832*x^9 - 141505556545693615275/8796093022208*x^7 + 4911763119767877555/17592186044416*x^5 - 20146690401016725/8796093022208*x^3 + 395033145117975/70368744177664*x
50:
12611418068195524166851562157/140737488355328*x^50 - 156050375086257748529223875175/140737488355328*x^48 + 226836112238787036521861509275/35184372088832*x^46 - 823773249709279237895181270525/35184372088832*x^44 + 4189728463575151392735706892025/70368744177664*x^42 - 7928255400303748020099876118755/70368744177664*x^40 + 5790298887862287879848224131675/35184372088832*x^38 - 6684039602901787158511168414725/35184372088832*x^36 + 24770264410753681822717859419275/140737488355328*x^34 - 18602568051449552212241926551825/140737488355328*x^32 + 1423900270604780539702468452115/17592186044416*x^30 - 712769410486857922635873160725/17592186044416*x^28 + 583174972216520118520259858775/35184372088832*x^26 - 194391657405506706173419952925/35184372088832*x^24 + 26248579962778792027330678575/17592186044416*x^22 - 5693353963757653481984400705/17592186044416*x^20 + 7838675747202566388239392275/140737488355328*x^18 - 1052956443654076082002306425/140737488355328*x^16 + 26998883170617335435956575/35184372088832*x^14 - 2052546673789621992207225/35184372088832*x^12 + 222078820442811559812585/70368744177664*x^10 - 8065816723104536070675/70368744177664*x^8 + 90048990529077755175/35184372088832*x^6 - 1067774591253886425/35184372088832*x^4 + 20146690401016725/140737488355328*x^2 - 15801325804719/140737488355328
Definition
The Legendre polynomials $P_n$, $n\geq 0$, can be defined as the system of orthogonal polynomials with respect to the inner product (3) such that $\deg P_n=n$ and $P_n(1)=1$.
Parameters
$n$
—   integer ($n \geq 0$)
Formulas
(1)
$P_0(x) = 1$, $P_1(x) = x$, and $(n+1)P_{n+1}(x) = (2n+1)_x P_n(x) - n P_{n-1}(x)$ for $n\geq 1$ (Bonnet's recursion formula).
(2)
$\sum_{n=0}^\infty P_n(x)t^n = \frac{1}{\sqrt{1-2tx+t^2}}$ (generating function).
Comments
(3)
The $P_n$ are orthogonal with respect to the inner product $\langle f,g\rangle = \int_{-1}^1 f(x)g(x) dx$.
Programs
(P1)
Sage
polynomials = {n: legendre_P(n,x) for n in [0..100]}
Links
Data properties
Entries are of type: rational polynomial
Table is complete: no