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Gegenbauer polynomials
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Polynomials
$\alpha$
$n$ 
$C_n^{\alpha}(x)$
a
0:
1
a
1:
2*a*x
a
2:
2*(a + 1)*a*x^2 - a
a
3:
4/3*(a + 2)*(a + 1)*a*x^3 - 2*(a + 1)*a*x
a
4:
2/3*(a + 3)*(a + 2)*(a + 1)*a*x^4 - 2*(a + 2)*(a + 1)*a*x^2 + 1/2*(a + 1)*a
a
5:
4/15*(a + 4)*(a + 3)*(a + 2)*(a + 1)*a*x^5 - 4/3*(a + 3)*(a + 2)*(a + 1)*a*x^3 + (a + 2)*(a + 1)*a*x
a
6:
4/45*(a + 5)*(a + 4)*(a + 3)*(a + 2)*(a + 1)*a*x^6 - 2/3*(a + 4)*(a + 3)*(a + 2)*(a + 1)*a*x^4 + (a + 3)*(a + 2)*(a + 1)*a*x^2 - 1/6*(a + 2)*(a + 1)*a
a
7:
8/315*(a + 6)*(a + 5)*(a + 4)*(a + 3)*(a + 2)*(a + 1)*a*x^7 - 4/15*(a + 5)*(a + 4)*(a + 3)*(a + 2)*(a + 1)*a*x^5 + 2/3*(a + 4)*(a + 3)*(a + 2)*(a + 1)*a*x^3 - 1/3*(a + 3)*(a + 2)*(a + 1)*a*x
a
8:
2/315*(a + 7)*(a + 6)*(a + 5)*(a + 4)*(a + 3)*(a + 2)*(a + 1)*a*x^8 - 4/45*(a + 6)*(a + 5)*(a + 4)*(a + 3)*(a + 2)*(a + 1)*a*x^6 + 1/3*(a + 5)*(a + 4)*(a + 3)*(a + 2)*(a + 1)*a*x^4 - 1/3*(a + 4)*(a + 3)*(a + 2)*(a + 1)*a*x^2 + 1/24*(a + 3)*(a + 2)*(a + 1)*a
a
9:
4/2835*(a + 8)*(a + 7)*(a + 6)*(a + 5)*(a + 4)*(a + 3)*(a + 2)*(a + 1)*a*x^9 - 8/315*(a + 7)*(a + 6)*(a + 5)*(a + 4)*(a + 3)*(a + 2)*(a + 1)*a*x^7 + 2/15*(a + 6)*(a + 5)*(a + 4)*(a + 3)*(a + 2)*(a + 1)*a*x^5 - 2/9*(a + 5)*(a + 4)*(a + 3)*(a + 2)*(a + 1)*a*x^3 + 1/12*(a + 4)*(a + 3)*(a + 2)*(a + 1)*a*x
a
10:
4/14175*(a + 9)*(a + 8)*(a + 7)*(a + 6)*(a + 5)*(a + 4)*(a + 3)*(a + 2)*(a + 1)*a*x^10 - 2/315*(a + 8)*(a + 7)*(a + 6)*(a + 5)*(a + 4)*(a + 3)*(a + 2)*(a + 1)*a*x^8 + 2/45*(a + 7)*(a + 6)*(a + 5)*(a + 4)*(a + 3)*(a + 2)*(a + 1)*a*x^6 - 1/9*(a + 6)*(a + 5)*(a + 4)*(a + 3)*(a + 2)*(a + 1)*a*x^4 + 1/12*(a + 5)*(a + 4)*(a + 3)*(a + 2)*(a + 1)*a*x^2 - 1/120*(a + 4)*(a + 3)*(a + 2)*(a + 1)*a
a
11:
8/155925*(a + 10)*(a + 9)*(a + 8)*(a + 7)*(a + 6)*(a + 5)*(a + 4)*(a + 3)*(a + 2)*(a + 1)*a*x^11 - 4/2835*(a + 9)*(a + 8)*(a + 7)*(a + 6)*(a + 5)*(a + 4)*(a + 3)*(a + 2)*(a + 1)*a*x^9 + 4/315*(a + 8)*(a + 7)*(a + 6)*(a + 5)*(a + 4)*(a + 3)*(a + 2)*(a + 1)*a*x^7 - 2/45*(a + 7)*(a + 6)*(a + 5)*(a + 4)*(a + 3)*(a + 2)*(a + 1)*a*x^5 + 1/18*(a + 6)*(a + 5)*(a + 4)*(a + 3)*(a + 2)*(a + 1)*a*x^3 - 1/60*(a + 5)*(a + 4)*(a + 3)*(a + 2)*(a + 1)*a*x
a
12:
4/467775*(a + 11)*(a + 10)*(a + 9)*(a + 8)*(a + 7)*(a + 6)*(a + 5)*(a + 4)*(a + 3)*(a + 2)*(a + 1)*a*x^12 - 4/14175*(a + 10)*(a + 9)*(a + 8)*(a + 7)*(a + 6)*(a + 5)*(a + 4)*(a + 3)*(a + 2)*(a + 1)*a*x^10 + 1/315*(a + 9)*(a + 8)*(a + 7)*(a + 6)*(a + 5)*(a + 4)*(a + 3)*(a + 2)*(a + 1)*a*x^8 - 2/135*(a + 8)*(a + 7)*(a + 6)*(a + 5)*(a + 4)*(a + 3)*(a + 2)*(a + 1)*a*x^6 + 1/36*(a + 7)*(a + 6)*(a + 5)*(a + 4)*(a + 3)*(a + 2)*(a + 1)*a*x^4 - 1/60*(a + 6)*(a + 5)*(a + 4)*(a + 3)*(a + 2)*(a + 1)*a*x^2 + 1/720*(a + 5)*(a + 4)*(a + 3)*(a + 2)*(a + 1)*a
a
13:
8/6081075*(a + 12)*(a + 11)*(a + 10)*(a + 9)*(a + 8)*(a + 7)*(a + 6)*(a + 5)*(a + 4)*(a + 3)*(a + 2)*(a + 1)*a*x^13 - 8/155925*(a + 11)*(a + 10)*(a + 9)*(a + 8)*(a + 7)*(a + 6)*(a + 5)*(a + 4)*(a + 3)*(a + 2)*(a + 1)*a*x^11 + 2/2835*(a + 10)*(a + 9)*(a + 8)*(a + 7)*(a + 6)*(a + 5)*(a + 4)*(a + 3)*(a + 2)*(a + 1)*a*x^9 - 4/945*(a + 9)*(a + 8)*(a + 7)*(a + 6)*(a + 5)*(a + 4)*(a + 3)*(a + 2)*(a + 1)*a*x^7 + 1/90*(a + 8)*(a + 7)*(a + 6)*(a + 5)*(a + 4)*(a + 3)*(a + 2)*(a + 1)*a*x^5 - 1/90*(a + 7)*(a + 6)*(a + 5)*(a + 4)*(a + 3)*(a + 2)*(a + 1)*a*x^3 + 1/360*(a + 6)*(a + 5)*(a + 4)*(a + 3)*(a + 2)*(a + 1)*a*x
a
14:
8/42567525*(a + 13)*(a + 12)*(a + 11)*(a + 10)*(a + 9)*(a + 8)*(a + 7)*(a + 6)*(a + 5)*(a + 4)*(a + 3)*(a + 2)*(a + 1)*a*x^14 - 4/467775*(a + 12)*(a + 11)*(a + 10)*(a + 9)*(a + 8)*(a + 7)*(a + 6)*(a + 5)*(a + 4)*(a + 3)*(a + 2)*(a + 1)*a*x^12 + 2/14175*(a + 11)*(a + 10)*(a + 9)*(a + 8)*(a + 7)*(a + 6)*(a + 5)*(a + 4)*(a + 3)*(a + 2)*(a + 1)*a*x^10 - 1/945*(a + 10)*(a + 9)*(a + 8)*(a + 7)*(a + 6)*(a + 5)*(a + 4)*(a + 3)*(a + 2)*(a + 1)*a*x^8 + 1/270*(a + 9)*(a + 8)*(a + 7)*(a + 6)*(a + 5)*(a + 4)*(a + 3)*(a + 2)*(a + 1)*a*x^6 - 1/180*(a + 8)*(a + 7)*(a + 6)*(a + 5)*(a + 4)*(a + 3)*(a + 2)*(a + 1)*a*x^4 + 1/360*(a + 7)*(a + 6)*(a + 5)*(a + 4)*(a + 3)*(a + 2)*(a + 1)*a*x^2 - 1/5040*(a + 6)*(a + 5)*(a + 4)*(a + 3)*(a + 2)*(a + 1)*a
a
15:
16/638512875*(a + 14)*(a + 13)*(a + 12)*(a + 11)*(a + 10)*(a + 9)*(a + 8)*(a + 7)*(a + 6)*(a + 5)*(a + 4)*(a + 3)*(a + 2)*(a + 1)*a*x^15 - 8/6081075*(a + 13)*(a + 12)*(a + 11)*(a + 10)*(a + 9)*(a + 8)*(a + 7)*(a + 6)*(a + 5)*(a + 4)*(a + 3)*(a + 2)*(a + 1)*a*x^13 + 4/155925*(a + 12)*(a + 11)*(a + 10)*(a + 9)*(a + 8)*(a + 7)*(a + 6)*(a + 5)*(a + 4)*(a + 3)*(a + 2)*(a + 1)*a*x^11 - 2/8505*(a + 11)*(a + 10)*(a + 9)*(a + 8)*(a + 7)*(a + 6)*(a + 5)*(a + 4)*(a + 3)*(a + 2)*(a + 1)*a*x^9 + 1/945*(a + 10)*(a + 9)*(a + 8)*(a + 7)*(a + 6)*(a + 5)*(a + 4)*(a + 3)*(a + 2)*(a + 1)*a*x^7 - 1/450*(a + 9)*(a + 8)*(a + 7)*(a + 6)*(a + 5)*(a + 4)*(a + 3)*(a + 2)*(a + 1)*a*x^5 + 1/540*(a + 8)*(a + 7)*(a + 6)*(a + 5)*(a + 4)*(a + 3)*(a + 2)*(a + 1)*a*x^3 - 1/2520*(a + 7)*(a + 6)*(a + 5)*(a + 4)*(a + 3)*(a + 2)*(a + 1)*a*x
-1
0:
1
-1
1:
-2*x
-1
2:
1
-1
3:
0
-3/4
0:
1
-3/4
1:
-3/2*x
-3/4
2:
-3/8*x^2 + 3/4
-3/4
3:
-5/16*x^3 + 3/8*x
-3/4
4:
-45/128*x^4 + 15/32*x^2 - 3/32
-3/4
5:
-117/256*x^5 + 45/64*x^3 - 15/64*x
-3/4
6:
-663/1024*x^6 + 585/512*x^4 - 135/256*x^2 + 5/128
-3/4
7:
-1989/2048*x^7 + 1989/1024*x^5 - 585/512*x^3 + 45/256*x
-3/4
8:
-49725/32768*x^8 + 13923/4096*x^6 - 9945/4096*x^4 + 585/1024*x^2 - 45/2048
-3/4
9:
-160225/65536*x^9 + 49725/8192*x^7 - 41769/8192*x^5 + 3315/2048*x^3 - 585/4096*x
-3/4
10:
-1057485/262144*x^10 + 1442025/131072*x^8 - 348075/32768*x^6 + 69615/16384*x^4 - 9945/16384*x^2 + 117/8192
-3/4
11:
-3556995/524288*x^11 + 5287425/262144*x^9 - 1442025/65536*x^7 + 348075/32768*x^5 - 69615/32768*x^3 + 1989/16384*x
-3/4
12:
-48612265/4194304*x^12 + 39126945/1048576*x^10 - 47586825/1048576*x^8 + 3364725/131072*x^6 - 1740375/262144*x^4 + 41769/65536*x^2 - 663/65536
-3/4
13:
-168273225/8388608*x^13 + 145836795/2097152*x^11 - 195634725/2097152*x^9 + 15862275/262144*x^7 - 10094175/524288*x^5 + 348075/131072*x^3 - 13923/131072*x
-3/4
14:
-1177912575/33554432*x^14 + 2187551925/16777216*x^12 - 1604204745/8388608*x^10 + 586904175/4194304*x^8 - 111035925/2097152*x^6 + 10094175/1048576*x^4 - 348075/524288*x^2 + 1989/262144
-3/4
15:
-4161957765/67108864*x^15 + 8245388025/33554432*x^13 - 6562655775/16777216*x^11 + 2673674575/8388608*x^9 - 586904175/4194304*x^7 + 66621555/2097152*x^5 - 3364725/1048576*x^3 + 49725/524288*x
-3/4
16:
-237231592605/2147483648*x^16 + 62429366475/134217728*x^14 - 107190044325/134217728*x^12 + 24063071175/33554432*x^10 - 24063071175/67108864*x^8 + 821665845/8388608*x^6 - 111035925/8388608*x^4 + 1442025/2097152*x^2 - 49725/8388608
-3/4
17:
-851242773465/4294967296*x^17 + 237231592605/268435456*x^15 - 437005565325/268435456*x^13 + 107190044325/67108864*x^11 - 120315355875/134217728*x^9 + 4812614235/16777216*x^7 - 821665845/16777216*x^5 + 15862275/4194304*x^3 - 1442025/16777216*x
-3/4
18:
-6147864475025/17179869184*x^18 + 14471127148905/8589934592*x^16 - 3558473889075/1073741824*x^14 + 1893690783075/536870912*x^12 - 1179090487575/536870912*x^10 + 216567640575/268435456*x^8 - 11229433215/67108864*x^6 + 586904175/33554432*x^4 - 47586825/67108864*x^2 + 160225/33554432
-3/4
19:
-22326455198775/34359738368*x^19 + 55330780275225/17179869184*x^17 - 14471127148905/2147483648*x^15 + 8303105741175/1073741824*x^13 - 5681072349225/1073741824*x^11 + 1179090487575/536870912*x^9 - 72189213525/134217728*x^7 + 4812614235/67108864*x^5 - 586904175/134217728*x^3 + 5287425/67108864*x
-3/4
20:
-325966245902115/274877906944*x^20 + 424202648776725/68719476736*x^18 - 940623264678825/68719476736*x^16 + 72355635744525/4294967296*x^14 - 107940374635275/8589934592*x^12 + 12498359168295/2147483648*x^10 - 3537271462725/2147483648*x^8 + 72189213525/268435456*x^6 - 24063071175/1073741824*x^4 + 195634725/268435456*x^2 - 1057485/268435456
-2/3
0:
1
-2/3
1:
-4/3*x
-2/3
2:
-4/9*x^2 + 2/3
-2/3
3:
-32/81*x^3 + 4/9*x
-2/3
4:
-112/243*x^4 + 16/27*x^2 - 1/9
-2/3
5:
-448/729*x^5 + 224/243*x^3 - 8/27*x
-2/3
6:
-5824/6561*x^6 + 1120/729*x^4 - 56/81*x^2 + 4/81
-2/3
7:
-26624/19683*x^7 + 5824/2187*x^5 - 1120/729*x^3 + 56/243*x
-2/3
8:
-126464/59049*x^8 + 93184/19683*x^6 - 7280/2187*x^4 + 560/729*x^2 - 7/243
-2/3
9:
-5564416/1594323*x^9 + 505856/59049*x^7 - 46592/6561*x^5 + 14560/6561*x^3 - 140/729*x
-2/3
10:
-27822080/4782969*x^10 + 2782208/177147*x^8 - 885248/59049*x^6 + 116480/19683*x^4 - 1820/2187*x^2 + 14/729
-2/3
11:
-141639680/14348907*x^11 + 139110400/4782969*x^9 - 5564416/177147*x^7 + 885248/59049*x^5 - 58240/19683*x^3 + 364/2187*x
-2/3
12:
-2195415040/129140163*x^12 + 779018240/14348907*x^10 - 34777600/531441*x^8 + 19475456/531441*x^6 - 553280/59049*x^4 + 5824/6561*x^2 - 91/6561
-2/3
13:
-11483709440/387420489*x^13 + 4390830080/43046721*x^11 - 1947545600/14348907*x^9 + 139110400/1594323*x^7 - 4868864/177147*x^5 + 221312/59049*x^3 - 2912/19683*x
-2/3
14:
-60699607040/1162261467*x^14 + 74644111360/387420489*x^12 - 12074782720/43046721*x^10 + 973772800/4782969*x^8 - 121721600/1594323*x^6 + 2434432/177147*x^4 - 55328/59049*x^2 + 208/19683
-2/3
15:
-971193712640/10460353203*x^15 + 424897249280/1162261467*x^13 - 74644111360/129140163*x^11 + 60373913600/129140163*x^9 - 973772800/4782969*x^7 + 24344320/531441*x^5 - 2434432/531441*x^3 + 7904/59049*x
-2/3
16:
-5220166205440/31381059609*x^16 + 2427984281600/3486784401*x^14 - 1380916060160/1162261467*x^12 + 410542612480/387420489*x^10 - 7546739200/14348907*x^8 + 681640960/4782969*x^6 - 30430400/1594323*x^4 + 173888/177147*x^2 - 494/59049
-2/3
17:
-28250311229440/94143178827*x^17 + 41761329643520/31381059609*x^15 - 8497944985600/3486784401*x^13 + 2761832120320/1162261467*x^11 - 513178265600/387420489*x^9 + 6037391360/14348907*x^7 - 340820480/4782969*x^5 + 8694400/1594323*x^3 - 21736/177147*x
-2/3
18:
-1384265250242560/2541865828329*x^18 + 240127645450240/94143178827*x^16 - 52201662054400/10460353203*x^14 + 55236642406400/10460353203*x^12 - 3797519165440/1162261467*x^10 + 51317826560/43046721*x^8 - 10565434880/43046721*x^6 + 121721600/4782969*x^4 - 543400/531441*x^2 + 10868/1594323
-2/3
19:
-7577030843432960/7625597484987*x^19 + 1384265250242560/282429536481*x^17 - 960510581800960/94143178827*x^15 + 365411634380800/31381059609*x^13 - 27618321203200/3486784401*x^11 + 3797519165440/1162261467*x^9 - 102635653120/129140163*x^7 + 1509347840/14348907*x^5 - 30430400/4782969*x^3 + 543400/4782969*x
-2/3
20:
-41673669638881280/22876792454961*x^20 + 71981793012613120/7625597484987*x^18 - 5883127313530880/282429536481*x^16 + 2401276454502400/94143178827*x^14 - 593793905868800/31381059609*x^12 + 30380153323520/3486784401*x^10 - 949379791360/387420489*x^8 + 51317826560/129140163*x^6 - 471671200/14348907*x^4 + 15215200/14348907*x^2 - 27170/4782969
-1/2
0:
1
-1/2
1:
-x
-1/2
2:
-1/2*x^2 + 1/2
-1/2
3:
-1/2*x^3 + 1/2*x
-1/2
4:
-5/8*x^4 + 3/4*x^2 - 1/8
-1/2
5:
-7/8*x^5 + 5/4*x^3 - 3/8*x
-1/2
6:
-21/16*x^6 + 35/16*x^4 - 15/16*x^2 + 1/16
-1/2
7:
-33/16*x^7 + 63/16*x^5 - 35/16*x^3 + 5/16*x
-1/2
8:
-429/128*x^8 + 231/32*x^6 - 315/64*x^4 + 35/32*x^2 - 5/128
-1/2
9:
-715/128*x^9 + 429/32*x^7 - 693/64*x^5 + 105/32*x^3 - 35/128*x
-1/2
10:
-2431/256*x^10 + 6435/256*x^8 - 3003/128*x^6 + 1155/128*x^4 - 315/256*x^2 + 7/256
-1/2
11:
-4199/256*x^11 + 12155/256*x^9 - 6435/128*x^7 + 3003/128*x^5 - 1155/256*x^3 + 63/256*x
-1/2
12:
-29393/1024*x^12 + 46189/512*x^10 - 109395/1024*x^8 + 15015/256*x^6 - 15015/1024*x^4 + 693/512*x^2 - 21/1024
-1/2
13:
-52003/1024*x^13 + 88179/512*x^11 - 230945/1024*x^9 + 36465/256*x^7 - 45045/1024*x^5 + 3003/512*x^3 - 231/1024*x
-1/2
14:
-185725/2048*x^14 + 676039/2048*x^12 - 969969/2048*x^10 + 692835/2048*x^8 - 255255/2048*x^6 + 45045/2048*x^4 - 3003/2048*x^2 + 33/2048
-1/2
15:
-334305/2048*x^15 + 1300075/2048*x^13 - 2028117/2048*x^11 + 1616615/2048*x^9 - 692835/2048*x^7 + 153153/2048*x^5 - 15015/2048*x^3 + 429/2048*x
-1/2
16:
-9694845/32768*x^16 + 5014575/4096*x^14 - 16900975/8192*x^12 + 7436429/4096*x^10 - 14549535/16384*x^8 + 969969/4096*x^6 - 255255/8192*x^4 + 6435/4096*x^2 - 429/32768
-1/2
17:
-17678835/32768*x^17 + 9694845/4096*x^15 - 35102025/8192*x^13 + 16900975/4096*x^11 - 37182145/16384*x^9 + 2909907/4096*x^7 - 969969/8192*x^5 + 36465/4096*x^3 - 6435/32768*x
-1/2
18:
-64822395/65536*x^18 + 300540195/65536*x^16 - 145422675/16384*x^14 + 152108775/16384*x^12 - 185910725/32768*x^10 + 66927861/32768*x^8 - 6789783/16384*x^6 + 692835/16384*x^4 - 109395/65536*x^2 + 715/65536
-1/2
19:
-119409675/65536*x^19 + 583401555/65536*x^17 - 300540195/16384*x^15 + 339319575/16384*x^13 - 456326325/32768*x^11 + 185910725/32768*x^9 - 22309287/16384*x^7 + 2909907/16384*x^5 - 692835/65536*x^3 + 12155/65536*x
-1/2
20:
-883631595/262144*x^20 + 2268783825/131072*x^18 - 9917826435/262144*x^16 + 1502700975/32768*x^14 - 4411154475/131072*x^12 + 1003917915/65536*x^10 - 557732175/131072*x^8 + 22309287/32768*x^6 - 14549535/262144*x^4 + 230945/131072*x^2 - 2431/262144
-1/3
0:
1
-1/3
1:
-2/3*x
-1/3
2:
-4/9*x^2 + 1/3
-1/3
3:
-40/81*x^3 + 4/9*x
-1/3
4:
-160/243*x^4 + 20/27*x^2 - 1/9
-1/3
5:
-704/729*x^5 + 320/243*x^3 - 10/27*x
-1/3
6:
-9856/6561*x^6 + 1760/729*x^4 - 80/81*x^2 + 5/81
-1/3
7:
-47872/19683*x^7 + 9856/2187*x^5 - 1760/729*x^3 + 80/243*x
-1/3
8:
-239360/59049*x^8 + 167552/19683*x^6 - 12320/2187*x^4 + 880/729*x^2 - 10/243
-1/3
9:
-11010560/1594323*x^9 + 957440/59049*x^7 - 83776/6561*x^5 + 24640/6561*x^3 - 220/729*x
-1/3
10:
-57254912/4782969*x^10 + 5505280/177147*x^8 - 1675520/59049*x^6 + 209440/19683*x^4 - 3080/2187*x^2 + 22/729
-1/3
11:
-301889536/14348907*x^11 + 286274560/4782969*x^9 - 11010560/177147*x^7 + 1675520/59049*x^5 - 104720/19683*x^3 + 616/2187*x
-1/3
12:
-4830232576/129140163*x^12 + 1660392448/14348907*x^10 - 71568640/531441*x^8 + 38536960/531441*x^6 - 1047200/59049*x^4 + 10472/6561*x^2 - 154/6561
-1/3
13:
-26008944640/387420489*x^13 + 9660465152/43046721*x^11 - 4150981120/14348907*x^9 + 286274560/1594323*x^7 - 9634240/177147*x^5 + 418880/59049*x^3 - 5236/19683*x
-1/3
14:
-141191413760/1162261467*x^14 + 169058140160/387420489*x^12 - 26566279168/43046721*x^10 + 2075490560/4782969*x^8 - 250490240/1594323*x^6 + 4817120/177147*x^4 - 104720/59049*x^2 + 374/19683
-1/3
15:
-2315539185664/10460353203*x^15 + 988339896320/1162261467*x^13 - 169058140160/129140163*x^11 + 132831395840/129140163*x^9 - 2075490560/4782969*x^7 + 50098048/531441*x^5 - 4817120/531441*x^3 + 14960/59049*x
-1/3
16:
-12735465521152/31381059609*x^16 + 5788847964160/3486784401*x^14 - 3212104663040/1162261467*x^12 + 929819770880/387420489*x^10 - 16603924480/14348907*x^8 + 1452843392/4782969*x^6 - 62622560/1594323*x^4 + 344080/177147*x^2 - 935/59049
-1/3
17:
-70419632881664/94143178827*x^17 + 101883724169216/31381059609*x^15 - 20260967874560/3486784401*x^13 + 6424209326080/1162261467*x^11 - 1162274713600/387420489*x^9 + 13283139584/14348907*x^7 - 726421696/4782969*x^5 + 17892160/1594323*x^3 - 43010/177147*x
-1/3
18:
-3520981644083200/2541865828329*x^18 + 598566879494144/94143178827*x^16 - 127354655211520/10460353203*x^14 + 131696291184640/10460353203*x^12 - 8833287823360/1162261467*x^10 + 116227471360/43046721*x^8 - 23245494272/43046721*x^6 + 259436320/4782969*x^4 - 1118260/531441*x^2 + 21505/1594323
-1/3
19:
-19643371277516800/7625597484987*x^19 + 3520981644083200/282429536481*x^17 - 2394267517976576/94143178827*x^15 + 891482586480640/31381059609*x^13 - 65848145592320/3486784401*x^11 + 8833287823360/1162261467*x^9 - 232454942720/129140163*x^7 + 3320784896/14348907*x^5 - 64859080/4782969*x^3 + 1118260/4782969*x
-1/3
20:
-110002879154094080/22876792454961*x^20 + 186612027136409600/7625597484987*x^18 - 14964171987353600/282429536481*x^16 + 5985668794941440/94143178827*x^14 - 1448659203031040/31381059609*x^12 + 72432960151552/3486784401*x^10 - 2208321955840/387420489*x^8 + 116227471360/129140163*x^6 - 1037745280/14348907*x^4 + 32429540/14348907*x^2 - 55913/4782969
-1/4
0:
1
-1/4
1:
-1/2*x
-1/4
2:
-3/8*x^2 + 1/4
-1/4
3:
-7/16*x^3 + 3/8*x
-1/4
4:
-77/128*x^4 + 21/32*x^2 - 3/32
-1/4
5:
-231/256*x^5 + 77/64*x^3 - 21/64*x
-1/4
6:
-1463/1024*x^6 + 1155/512*x^4 - 231/256*x^2 + 7/128
-1/4
7:
-4807/2048*x^7 + 4389/1024*x^5 - 1155/512*x^3 + 77/256*x
-1/4
8:
-129789/32768*x^8 + 33649/4096*x^6 - 21945/4096*x^4 + 1155/1024*x^2 - 77/2048
-1/4
9:
-447051/65536*x^9 + 129789/8192*x^7 - 100947/8192*x^5 + 7315/2048*x^3 - 1155/4096*x
-1/4
10:
-3129357/262144*x^10 + 4023459/131072*x^8 - 908523/32768*x^6 + 168245/16384*x^4 - 21945/16384*x^2 + 231/8192
-1/4
11:
-11094993/524288*x^11 + 15646785/262144*x^9 - 4023459/65536*x^7 + 908523/32768*x^5 - 168245/32768*x^3 + 4389/16384*x
-1/4
12:
-159028233/4194304*x^12 + 122044923/1048576*x^10 - 140821065/1048576*x^8 + 9388071/131072*x^6 - 4542615/262144*x^4 + 100947/65536*x^2 - 1463/65536
-1/4
13:
-574948227/8388608*x^13 + 477084699/2097152*x^11 - 610224615/2097152*x^9 + 46940355/262144*x^7 - 28164213/524288*x^5 + 908523/131072*x^3 - 33649/131072*x
-1/4
14:
-4188908511/33554432*x^14 + 7474326951/16777216*x^12 - 5247931689/8388608*x^10 + 1830673845/4194304*x^8 - 328582485/2097152*x^6 + 28164213/1048576*x^4 - 908523/524288*x^2 + 4807/262144
-1/4
15:
-15359331207/67108864*x^15 + 29322359577/33554432*x^13 - 22422980853/16777216*x^11 + 8746552815/8388608*x^9 - 1830673845/4194304*x^7 + 197149491/2097152*x^5 - 9388071/1048576*x^3 + 129789/524288*x
-1/4
16:
-906200541213/2147483648*x^16 + 230389968105/134217728*x^14 - 381190674501/134217728*x^12 + 82217596461/33554432*x^10 - 78718975335/67108864*x^8 + 2562943383/8388608*x^6 - 328582485/8388608*x^4 + 4023459/2097152*x^2 - 129789/8388608
-1/4
17:
-3358272593907/4294967296*x^17 + 906200541213/268435456*x^15 - 1612729776735/268435456*x^13 + 381190674501/67108864*x^11 - 411087982305/134217728*x^9 + 15743795067/16777216*x^7 - 2562943383/16777216*x^5 + 46940355/4194304*x^3 - 4023459/16777216*x
-1/4
18:
-25000473754641/17179869184*x^18 + 57090634096419/8589934592*x^16 - 13593008118195/1073741824*x^14 + 6988495699185/536870912*x^12 - 4193097419511/536870912*x^10 + 739958368149/268435456*x^8 - 36735521823/67108864*x^6 + 1830673845/33554432*x^4 - 140821065/67108864*x^2 + 447051/33554432
-1/4
19:
-93422822977869/34359738368*x^19 + 225004263791769/17179869184*x^17 - 57090634096419/2147483648*x^15 + 31717018942455/1073741824*x^13 - 20965487097555/1073741824*x^11 + 4193097419511/536870912*x^9 - 246652789383/134217728*x^7 + 15743795067/67108864*x^5 - 1830673845/134217728*x^3 + 15646785/67108864*x
-1/4
20:
-1401342344668035/274877906944*x^20 + 1775033636579511/68719476736*x^18 - 3825072484460073/68719476736*x^16 + 285453170482095/4294967296*x^14 - 412321246251915/8589934592*x^12 + 46124071614621/2147483648*x^10 - 12579292258533/2147483648*x^8 + 246652789383/268435456*x^6 - 78718975335/1073741824*x^4 + 610224615/268435456*x^2 - 3129357/268435456
0
0:
1
0
1:
0
1/4
0:
1
1/4
1:
1/2*x
1/4
2:
5/8*x^2 - 1/4
1/4
3:
15/16*x^3 - 5/8*x
1/4
4:
195/128*x^4 - 45/32*x^2 + 5/32
1/4
5:
663/256*x^5 - 195/64*x^3 + 45/64*x
1/4
6:
4641/1024*x^6 - 3315/512*x^4 + 585/256*x^2 - 15/128
1/4
7:
16575/2048*x^7 - 13923/1024*x^5 + 3315/512*x^3 - 195/256*x
1/4
8:
480675/32768*x^8 - 116025/4096*x^6 + 69615/4096*x^4 - 3315/1024*x^2 + 195/2048
1/4
9:
1762475/65536*x^9 - 480675/8192*x^7 + 348075/8192*x^5 - 23205/2048*x^3 + 3315/4096*x
1/4
10:
13042315/262144*x^10 - 15862275/131072*x^8 + 3364725/32768*x^6 - 580125/16384*x^4 + 69615/16384*x^2 - 663/8192
1/4
11:
48612265/524288*x^11 - 65211575/262144*x^9 + 15862275/65536*x^7 - 3364725/32768*x^5 + 580125/32768*x^3 - 13923/16384*x
1/4
12:
729183975/4194304*x^12 - 534734915/1048576*x^10 + 586904175/1048576*x^8 - 37011975/131072*x^6 + 16823625/262144*x^4 - 348075/65536*x^2 + 4641/65536
1/4
13:
2748462675/8388608*x^13 - 2187551925/2097152*x^11 + 2673674575/2097152*x^9 - 195634725/262144*x^7 + 111035925/524288*x^5 - 3364725/131072*x^3 + 116025/131072*x
1/4
14:
20809788825/33554432*x^14 - 35730014775/16777216*x^12 + 24063071175/8388608*x^10 - 8021023725/4194304*x^8 + 1369443075/2097152*x^6 - 111035925/1048576*x^4 + 3364725/524288*x^2 - 16575/262144
1/4
15:
79077197535/67108864*x^15 - 145668521775/33554432*x^13 + 107190044325/16777216*x^11 - 40105118625/8388608*x^9 + 8021023725/4194304*x^7 - 821665845/2097152*x^5 + 37011975/1048576*x^3 - 480675/524288*x
1/4
16:
4823709049635/2147483648*x^16 - 1186157963025/134217728*x^14 + 1893690783075/134217728*x^12 - 393030162525/33554432*x^10 + 360946067625/67108864*x^8 - 11229433215/8388608*x^6 + 1369443075/8388608*x^4 - 15862275/2097152*x^2 + 480675/8388608
1/4
17:
18443593425075/4294967296*x^17 - 4823709049635/268435456*x^15 + 8303105741175/268435456*x^13 - 1893690783075/67108864*x^11 + 1965150812625/134217728*x^9 - 72189213525/16777216*x^7 + 11229433215/16777216*x^5 - 195634725/4194304*x^3 + 15862275/16777216*x
1/4
18:
141400882925575/17179869184*x^18 - 313541088226275/8589934592*x^16 + 72355635744525/1073741824*x^14 - 35980124878425/536870912*x^12 + 20830598613825/536870912*x^10 - 3537271462725/268435456*x^8 + 168441498225/67108864*x^6 - 8021023725/33554432*x^4 + 586904175/67108864*x^2 - 1762475/33554432
1/4
19:
543277076503525/34359738368*x^19 - 1272607946330175/17179869184*x^17 + 313541088226275/2147483648*x^15 - 168829816737225/1073741824*x^13 + 107940374635275/1073741824*x^11 - 20830598613825/536870912*x^9 + 1179090487575/134217728*x^7 - 72189213525/67108864*x^5 + 8021023725/134217728*x^3 - 65211575/67108864*x
1/4
20:
8366466978154285/274877906944*x^20 - 10322264453566975/68719476736*x^18 + 21634335087612975/68719476736*x^16 - 1567705441131375/4294967296*x^14 + 2194787617583925/8589934592*x^12 - 237468824197605/2147483648*x^10 + 62491795841475/2147483648*x^8 - 1179090487575/268435456*x^6 + 360946067625/1073741824*x^4 - 2673674575/268435456*x^2 + 13042315/268435456
1/3
0:
1
1/3
1:
2/3*x
1/3
2:
8/9*x^2 - 1/3
1/3
3:
112/81*x^3 - 8/9*x
1/3
4:
560/243*x^4 - 56/27*x^2 + 2/9
1/3
5:
2912/729*x^5 - 1120/243*x^3 + 28/27*x
1/3
6:
46592/6561*x^6 - 7280/729*x^4 + 280/81*x^2 - 14/81
1/3
7:
252928/19683*x^7 - 46592/2187*x^5 + 7280/729*x^3 - 280/243*x
1/3
8:
1391104/59049*x^8 - 885248/19683*x^6 + 58240/2187*x^4 - 3640/729*x^2 + 35/243
1/3
9:
69555200/1594323*x^9 - 5564416/59049*x^7 + 442624/6561*x^5 - 116480/6561*x^3 + 910/729*x
1/3
10:
389509120/4782969*x^10 - 34777600/177147*x^8 + 9737728/59049*x^6 - 1106560/19683*x^4 + 14560/2187*x^2 - 91/729
1/3
11:
2195415040/14348907*x^11 - 1947545600/4782969*x^9 + 69555200/177147*x^7 - 9737728/59049*x^5 + 553280/19683*x^3 - 2912/2187*x
1/3
12:
37322055680/129140163*x^12 - 12074782720/14348907*x^10 + 486886400/531441*x^8 - 243443200/531441*x^6 + 6086080/59049*x^4 - 55328/6561*x^2 + 728/6561
1/3
13:
212448624640/387420489*x^13 - 74644111360/43046721*x^11 + 30186956800/14348907*x^9 - 1947545600/1594323*x^7 + 60860800/177147*x^5 - 2434432/59049*x^3 + 27664/19683*x
1/3
14:
1213992140800/1162261467*x^14 - 1380916060160/387420489*x^12 + 205271306240/43046721*x^10 - 15093478400/4782969*x^8 + 1704102400/1594323*x^6 - 30430400/177147*x^4 + 608608/59049*x^2 - 1976/19683
1/3
15:
20880664821760/10460353203*x^15 - 8497944985600/1162261467*x^13 + 1380916060160/129140163*x^11 - 1026356531200/129140163*x^9 + 15093478400/4782969*x^7 - 340820480/531441*x^5 + 30430400/531441*x^3 - 86944/59049*x
1/3
16:
120063822725120/31381059609*x^16 - 52201662054400/3486784401*x^14 + 27618321203200/1162261467*x^12 - 7595038330880/387420489*x^10 + 128294566400/14348907*x^8 - 10565434880/4782969*x^6 + 426025600/1594323*x^4 - 2173600/177147*x^2 + 5434/59049
1/3
17:
692132625121280/94143178827*x^17 - 960510581800960/31381059609*x^15 + 182705817190400/3486784401*x^13 - 55236642406400/1162261467*x^11 + 9493797913600/387420489*x^9 - 102635653120/14348907*x^7 + 5282717440/4782969*x^5 - 121721600/1594323*x^3 + 271700/177147*x
1/3
18:
35990896506306560/2541865828329*x^18 - 5883127313530880/94143178827*x^16 + 1200638227251200/10460353203*x^14 - 1187587811737600/10460353203*x^12 + 75950383308800/1162261467*x^10 - 949379791360/43046721*x^8 + 179612392960/43046721*x^6 - 1886684800/4782969*x^4 + 7607600/531441*x^2 - 135850/1594323
1/3
19:
208368348194406400/7625597484987*x^19 - 35990896506306560/282429536481*x^17 + 23532509254123520/94143178827*x^15 - 8404467590758400/31381059609*x^13 + 593793905868800/3486784401*x^11 - 75950383308800/1162261467*x^9 + 1898759582720/129140163*x^7 - 25658913280/14348907*x^5 + 471671200/4782969*x^3 - 7607600/4782969*x
1/3
20:
1208536419527557120/22876792454961*x^20 - 1979499307846860800/7625597484987*x^18 + 152961310151802880/282429536481*x^16 - 58831273135308800/94143178827*x^14 + 13657259834982400/31381059609*x^12 - 653173296455680/3486784401*x^10 + 18987595827200/387420489*x^8 - 949379791360/129140163*x^6 + 8018410400/14348907*x^4 - 235835600/14348907*x^2 + 380380/4782969
1/2:
equals: Legendre_polynomials
2/3
0:
1
2/3
1:
4/3*x
2/3
2:
20/9*x^2 - 2/3
2/3
3:
320/81*x^3 - 20/9*x
2/3
4:
1760/243*x^4 - 160/27*x^2 + 5/9
2/3
5:
9856/729*x^5 - 3520/243*x^3 + 80/27*x
2/3
6:
167552/6561*x^6 - 24640/729*x^4 + 880/81*x^2 - 40/81
2/3
7:
957440/19683*x^7 - 167552/2187*x^5 + 24640/729*x^3 - 880/243*x
2/3
8:
5505280/59049*x^8 - 3351040/19683*x^6 + 209440/2187*x^4 - 12320/729*x^2 + 110/243
2/3
9:
286274560/1594323*x^9 - 22021120/59049*x^7 + 1675520/6561*x^5 - 418880/6561*x^3 + 3080/729*x
2/3
10:
1660392448/4782969*x^10 - 143137280/177147*x^8 + 38536960/59049*x^6 - 4188800/19683*x^4 + 52360/2187*x^2 - 308/729
2/3
11:
9660465152/14348907*x^11 - 8301962240/4782969*x^9 + 286274560/177147*x^7 - 38536960/59049*x^5 + 2094400/19683*x^3 - 10472/2187*x
2/3
12:
169058140160/129140163*x^12 - 53132558336/14348907*x^10 + 2075490560/531441*x^8 - 1001960960/531441*x^6 + 24085600/59049*x^4 - 209440/6561*x^2 + 2618/6561
2/3
13:
988339896320/387420489*x^13 - 338116280320/43046721*x^11 + 132831395840/14348907*x^9 - 8301962240/1594323*x^7 + 250490240/177147*x^5 - 9634240/59049*x^3 + 104720/19683*x
2/3
14:
5788847964160/1162261467*x^14 - 6424209326080/387420489*x^12 + 929819770880/43046721*x^10 - 66415697920/4782969*x^8 + 7264216960/1594323*x^6 - 125245120/177147*x^4 + 2408560/59049*x^2 - 7480/19683
2/3
15:
101883724169216/10460353203*x^15 - 40521935749120/1162261467*x^13 + 6424209326080/129140163*x^11 - 4649098854400/129140163*x^9 + 66415697920/4782969*x^7 - 1452843392/531441*x^5 + 125245120/531441*x^3 - 344080/59049*x
2/3
16:
598566879494144/31381059609*x^16 - 254709310423040/3486784401*x^14 + 131696291184640/1162261467*x^12 - 35333151293440/387420489*x^10 + 581137356800/14348907*x^8 - 46490988544/4782969*x^6 + 1816054240/1594323*x^4 - 8946080/177147*x^2 + 21505/59049
2/3
17:
3520981644083200/94143178827*x^17 - 4788535035953152/31381059609*x^15 + 891482586480640/3486784401*x^13 - 263392582369280/1162261467*x^11 + 44166439116800/387420489*x^9 - 464909885440/14348907*x^7 + 23245494272/4782969*x^5 - 518872640/1594323*x^3 + 1118260/177147*x
2/3
18:
186612027136409600/2541865828329*x^18 - 29928343974707200/94143178827*x^16 + 5985668794941440/10460353203*x^14 - 5794636812124160/10460353203*x^12 + 362164800757760/1162261467*x^10 - 4416643911680/43046721*x^8 + 813592299520/43046721*x^6 - 8301962240/4782969*x^4 + 32429540/531441*x^2 - 559130/1594323
2/3
19:
1100028791540940800/7625597484987*x^19 - 186612027136409600/282429536481*x^17 + 119713375898828800/94143178827*x^15 - 41899681564590080/31381059609*x^13 + 2897318406062080/3486784401*x^11 - 362164800757760/1162261467*x^9 + 8833287823360/129140163*x^7 - 116227471360/14348907*x^5 + 2075490560/4782969*x^3 - 32429540/4782969*x
2/3
20:
6490169870091550720/22876792454961*x^20 - 10450273519638937600/7625597484987*x^18 + 793101115329740800/282429536481*x^16 - 299283439747072000/94143178827*x^14 + 68086982542458880/31381059609*x^12 - 3187050246668288/3486784401*x^10 + 90541200189440/387420489*x^8 - 4416643911680/129140163*x^6 + 36321084800/14348907*x^4 - 1037745280/14348907*x^2 + 1621477/4782969
3/4
0:
1
3/4
1:
3/2*x
3/4
2:
21/8*x^2 - 3/4
3/4
3:
77/16*x^3 - 21/8*x
3/4
4:
1155/128*x^4 - 231/32*x^2 + 21/32
3/4
5:
4389/256*x^5 - 1155/64*x^3 + 231/64*x
3/4
6:
33649/1024*x^6 - 21945/512*x^4 + 3465/256*x^2 - 77/128
3/4
7:
129789/2048*x^7 - 100947/1024*x^5 + 21945/512*x^3 - 1155/256*x
3/4
8:
4023459/32768*x^8 - 908523/4096*x^6 + 504735/4096*x^4 - 21945/1024*x^2 + 1155/2048
3/4
9:
15646785/65536*x^9 - 4023459/8192*x^7 + 2725569/8192*x^5 - 168245/2048*x^3 + 21945/4096*x
3/4
10:
122044923/262144*x^10 - 140821065/131072*x^8 + 28164213/32768*x^6 - 4542615/16384*x^4 + 504735/16384*x^2 - 4389/8192
3/4
11:
477084699/524288*x^11 - 610224615/262144*x^9 + 140821065/65536*x^7 - 28164213/32768*x^5 + 4542615/32768*x^3 - 100947/16384*x
3/4
12:
7474326951/4194304*x^12 - 5247931689/1048576*x^10 + 5492021535/1048576*x^8 - 328582485/131072*x^6 + 140821065/262144*x^4 - 2725569/65536*x^2 + 33649/65536
3/4
13:
29322359577/8388608*x^13 - 22422980853/2097152*x^11 + 26239658445/2097152*x^9 - 1830673845/262144*x^7 + 985747455/524288*x^5 - 28164213/131072*x^3 + 908523/131072*x
3/4
14:
230389968105/33554432*x^14 - 381190674501/16777216*x^12 + 246652789383/8388608*x^10 - 78718975335/4194304*x^8 + 12814716915/2097152*x^6 - 985747455/1048576*x^4 + 28164213/524288*x^2 - 129789/262144
3/4
15:
906200541213/67108864*x^15 - 1612729776735/33554432*x^13 + 1143572023503/16777216*x^11 - 411087982305/8388608*x^9 + 78718975335/4194304*x^7 - 7688830149/2097152*x^5 + 328582485/1048576*x^3 - 4023459/524288*x
3/4
16:
57090634096419/2147483648*x^16 - 13593008118195/134217728*x^14 + 20965487097555/134217728*x^12 - 4193097419511/33554432*x^10 + 3699791840745/67108864*x^8 - 110206565469/8388608*x^6 + 12814716915/8388608*x^4 - 140821065/2097152*x^2 + 4023459/8388608
3/4
17:
225004263791769/4294967296*x^17 - 57090634096419/268435456*x^15 + 95151056827365/268435456*x^13 - 20965487097555/67108864*x^11 + 20965487097555/134217728*x^9 - 739958368149/16777216*x^7 + 110206565469/16777216*x^5 - 1830673845/4194304*x^3 + 140821065/16777216*x
3/4
18:
1775033636579511/17179869184*x^18 - 3825072484460073/8589934592*x^16 + 856359511446285/1073741824*x^14 - 412321246251915/536870912*x^12 + 230620358073105/536870912*x^10 - 37737876775599/268435456*x^8 + 1726569525681/67108864*x^6 - 78718975335/33554432*x^4 + 5492021535/67108864*x^2 - 15646785/33554432
3/4
19:
7006711723340175/34359738368*x^19 - 15975302729215599/17179869184*x^17 + 3825072484460073/2147483648*x^15 - 1998172193374665/1073741824*x^13 + 1236963738755745/1073741824*x^11 - 230620358073105/536870912*x^9 + 12579292258533/134217728*x^7 - 739958368149/67108864*x^5 + 78718975335/134217728*x^3 - 610224615/67108864*x
3/4
20:
110706045228774765/274877906944*x^20 - 133127522743463325/68719476736*x^18 + 271580146396665183/68719476736*x^16 - 19125362422300365/4294967296*x^14 + 25976238513870645/8589934592*x^12 - 2721320225262639/2147483648*x^10 + 691861074219315/2147483648*x^8 - 12579292258533/268435456*x^6 + 3699791840745/1073741824*x^4 - 26239658445/268435456*x^2 + 122044923/268435456
3/2
0:
1
3/2
1:
3*x
3/2
2:
15/2*x^2 - 3/2
3/2
3:
35/2*x^3 - 15/2*x
3/2
4:
315/8*x^4 - 105/4*x^2 + 15/8
3/2
5:
693/8*x^5 - 315/4*x^3 + 105/8*x
3/2
6:
3003/16*x^6 - 3465/16*x^4 + 945/16*x^2 - 35/16
3/2
7:
6435/16*x^7 - 9009/16*x^5 + 3465/16*x^3 - 315/16*x
3/2
8:
109395/128*x^8 - 45045/32*x^6 + 45045/64*x^4 - 3465/32*x^2 + 315/128
3/2
9:
230945/128*x^9 - 109395/32*x^7 + 135135/64*x^5 - 15015/32*x^3 + 3465/128*x
3/2
10:
969969/256*x^10 - 2078505/256*x^8 + 765765/128*x^6 - 225225/128*x^4 + 45045/256*x^2 - 693/256
3/2
11:
2028117/256*x^11 - 4849845/256*x^9 + 2078505/128*x^7 - 765765/128*x^5 + 225225/256*x^3 - 9009/256*x
3/2
12:
16900975/1024*x^12 - 22309287/512*x^10 + 43648605/1024*x^8 - 4849845/256*x^6 + 3828825/1024*x^4 - 135135/512*x^2 + 3003/1024
3/2
13:
35102025/1024*x^13 - 50702925/512*x^11 + 111546435/1024*x^9 - 14549535/256*x^7 + 14549535/1024*x^5 - 765765/512*x^3 + 45045/1024*x
3/2
14:
145422675/2048*x^14 - 456326325/2048*x^12 + 557732175/2048*x^10 - 334639305/2048*x^8 + 101846745/2048*x^6 - 14549535/2048*x^4 + 765765/2048*x^2 - 6435/2048
3/2
15:
300540195/2048*x^15 - 1017958725/2048*x^13 + 1368978975/2048*x^11 - 929553625/2048*x^9 + 334639305/2048*x^7 - 61108047/2048*x^5 + 4849845/2048*x^3 - 109395/2048*x
3/2
16:
9917826435/32768*x^16 - 4508102925/4096*x^14 + 13233463425/8192*x^12 - 5019589575/4096*x^10 + 8365982625/16384*x^8 - 468495027/4096*x^6 + 101846745/8192*x^4 - 2078505/4096*x^2 + 109395/32768
3/2
17:
20419054425/32768*x^17 - 9917826435/4096*x^15 + 31556720475/8192*x^13 - 13233463425/4096*x^11 + 25097947875/16384*x^9 - 1673196525/4096*x^7 + 468495027/8192*x^5 - 14549535/4096*x^3 + 2078505/32768*x
3/2
18:
83945001525/65536*x^18 - 347123925225/65536*x^16 + 148767396525/16384*x^14 - 136745788725/16384*x^12 + 145568097675/32768*x^10 - 45176306175/32768*x^8 + 3904125225/16384*x^6 - 334639305/16384*x^4 + 43648605/65536*x^2 - 230945/65536
3/2
19:
172308161025/65536*x^19 - 755505013725/65536*x^17 + 347123925225/16384*x^15 - 347123925225/16384*x^13 + 410237366175/32768*x^11 - 145568097675/32768*x^9 + 15058768725/16384*x^7 - 1673196525/16384*x^5 + 334639305/65536*x^3 - 4849845/65536*x
3/2
20:
1412926920405/262144*x^20 - 3273855059475/131072*x^18 + 12843585233325/262144*x^16 - 1735619626125/32768*x^14 + 4512611027925/131072*x^12 - 902522205585/65536*x^10 + 436704293025/131072*x^8 - 15058768725/32768*x^6 + 8365982625/262144*x^4 - 111546435/131072*x^2 + 969969/262144
2
0:
1
2
1:
4*x
2
2:
12*x^2 - 2
2
3:
32*x^3 - 12*x
2
4:
80*x^4 - 48*x^2 + 3
2
5:
192*x^5 - 160*x^3 + 24*x
2
6:
448*x^6 - 480*x^4 + 120*x^2 - 4
2
7:
1024*x^7 - 1344*x^5 + 480*x^3 - 40*x
2
8:
2304*x^8 - 3584*x^6 + 1680*x^4 - 240*x^2 + 5
2
9:
5120*x^9 - 9216*x^7 + 5376*x^5 - 1120*x^3 + 60*x
2
10:
11264*x^10 - 23040*x^8 + 16128*x^6 - 4480*x^4 + 420*x^2 - 6
2
11:
24576*x^11 - 56320*x^9 + 46080*x^7 - 16128*x^5 + 2240*x^3 - 84*x
2
12:
53248*x^12 - 135168*x^10 + 126720*x^8 - 53760*x^6 + 10080*x^4 - 672*x^2 + 7
2
13:
114688*x^13 - 319488*x^11 + 337920*x^9 - 168960*x^7 + 40320*x^5 - 4032*x^3 + 112*x
2
14:
245760*x^14 - 745472*x^12 + 878592*x^10 - 506880*x^8 + 147840*x^6 - 20160*x^4 + 1008*x^2 - 8
2
15:
524288*x^15 - 1720320*x^13 + 2236416*x^11 - 1464320*x^9 + 506880*x^7 - 88704*x^5 + 6720*x^3 - 144*x
2
16:
1114112*x^16 - 3932160*x^14 + 5591040*x^12 - 4100096*x^10 + 1647360*x^8 - 354816*x^6 + 36960*x^4 - 1440*x^2 + 9
2
17:
2359296*x^17 - 8912896*x^15 + 13762560*x^13 - 11182080*x^11 + 5125120*x^9 - 1317888*x^7 + 177408*x^5 - 10560*x^3 + 180*x
2
18:
4980736*x^18 - 20054016*x^16 + 33423360*x^14 - 29818880*x^12 + 15375360*x^10 - 4612608*x^8 + 768768*x^6 - 63360*x^4 + 1980*x^2 - 10
2
19:
10485760*x^19 - 44826624*x^17 + 80216064*x^15 - 77987840*x^13 + 44728320*x^11 - 15375360*x^9 + 3075072*x^7 - 329472*x^5 + 15840*x^3 - 220*x
2
20:
22020096*x^20 - 99614720*x^18 + 190513152*x^16 - 200540160*x^14 + 126730240*x^12 - 49201152*x^10 + 11531520*x^8 - 1537536*x^6 + 102960*x^4 - 2640*x^2 + 11
5/2
0:
1
5/2
1:
5*x
5/2
2:
35/2*x^2 - 5/2
5/2
3:
105/2*x^3 - 35/2*x
5/2
4:
1155/8*x^4 - 315/4*x^2 + 35/8
5/2
5:
3003/8*x^5 - 1155/4*x^3 + 315/8*x
5/2
6:
15015/16*x^6 - 15015/16*x^4 + 3465/16*x^2 - 105/16
5/2
7:
36465/16*x^7 - 45045/16*x^5 + 15015/16*x^3 - 1155/16*x
5/2
8:
692835/128*x^8 - 255255/32*x^6 + 225225/64*x^4 - 15015/32*x^2 + 1155/128
5/2
9:
1616615/128*x^9 - 692835/32*x^7 + 765765/64*x^5 - 75075/32*x^3 + 15015/128*x
5/2
10:
7436429/256*x^10 - 14549535/256*x^8 + 4849845/128*x^6 - 1276275/128*x^4 + 225225/256*x^2 - 3003/256
5/2
11:
16900975/256*x^11 - 37182145/256*x^9 + 14549535/128*x^7 - 4849845/128*x^5 + 1276275/256*x^3 - 45045/256*x
5/2
12:
152108775/1024*x^12 - 185910725/512*x^10 + 334639305/1024*x^8 - 33948915/256*x^6 + 24249225/1024*x^4 - 765765/512*x^2 + 15015/1024
5/2
13:
339319575/1024*x^13 - 456326325/512*x^11 + 929553625/1024*x^9 - 111546435/256*x^7 + 101846745/1024*x^5 - 4849845/512*x^3 + 255255/1024*x
5/2
14:
1502700975/2048*x^14 - 4411154475/2048*x^12 + 5019589575/2048*x^10 - 2788660875/2048*x^8 + 780825045/2048*x^6 - 101846745/2048*x^4 + 4849845/2048*x^2 - 36465/2048
5/2
15:
3305942145/2048*x^15 - 10518906825/2048*x^13 + 13233463425/2048*x^11 - 8365982625/2048*x^9 + 2788660875/2048*x^7 - 468495027/2048*x^5 + 33948915/2048*x^3 - 692835/2048*x
5/2
16:
115707975075/32768*x^16 - 49589132175/4096*x^14 + 136745788725/8192*x^12 - 48522699225/4096*x^10 + 75293843625/16384*x^8 - 3904125225/4096*x^6 + 780825045/8192*x^4 - 14549535/4096*x^2 + 692835/32768
5/2
17:
251835004575/32768*x^17 - 115707975075/4096*x^15 + 347123925225/8192*x^13 - 136745788725/4096*x^11 + 242613496125/16384*x^9 - 15058768725/4096*x^7 + 3904125225/8192*x^5 - 111546435/4096*x^3 + 14549535/32768*x
5/2
18:
1091285019825/65536*x^18 - 4281195077775/65536*x^16 + 1735619626125/16384*x^14 - 1504203675975/16384*x^12 + 1504203675975/32768*x^10 - 436704293025/32768*x^8 + 35137127025/16384*x^6 - 2788660875/16384*x^4 + 334639305/65536*x^2 - 1616615/65536
5/2
19:
2354878200675/65536*x^19 - 9821565178425/65536*x^17 + 4281195077775/16384*x^15 - 4049779127625/16384*x^13 + 4512611027925/32768*x^11 - 1504203675975/32768*x^9 + 145568097675/16384*x^7 - 15058768725/16384*x^5 + 2788660875/65536*x^3 - 37182145/65536*x
5/2
20:
20251952525805/262144*x^20 - 44742685812825/131072*x^18 + 166966608033225/262144*x^16 - 21405975388875/32768*x^14 + 52647128659125/131072*x^12 - 9927744261435/65536*x^10 + 4512611027925/131072*x^8 - 145568097675/32768*x^6 + 75293843625/262144*x^4 - 929553625/131072*x^2 + 7436429/262144
3
0:
1
3
1:
6*x
3
2:
24*x^2 - 3
3
3:
80*x^3 - 24*x
3
4:
240*x^4 - 120*x^2 + 6
3
5:
672*x^5 - 480*x^3 + 60*x
3
6:
1792*x^6 - 1680*x^4 + 360*x^2 - 10
3
7:
4608*x^7 - 5376*x^5 + 1680*x^3 - 120*x
3
8:
11520*x^8 - 16128*x^6 + 6720*x^4 - 840*x^2 + 15
3
9:
28160*x^9 - 46080*x^7 + 24192*x^5 - 4480*x^3 + 210*x
3
10:
67584*x^10 - 126720*x^8 + 80640*x^6 - 20160*x^4 + 1680*x^2 - 21
3
11:
159744*x^11 - 337920*x^9 + 253440*x^7 - 80640*x^5 + 10080*x^3 - 336*x
3
12:
372736*x^12 - 878592*x^10 + 760320*x^8 - 295680*x^6 + 50400*x^4 - 3024*x^2 + 28
3
13:
860160*x^13 - 2236416*x^11 + 2196480*x^9 - 1013760*x^7 + 221760*x^5 - 20160*x^3 + 504*x
3
14:
1966080*x^14 - 5591040*x^12 + 6150144*x^10 - 3294720*x^8 + 887040*x^6 - 110880*x^4 + 5040*x^2 - 36
3
15:
4456448*x^15 - 13762560*x^13 + 16773120*x^11 - 10250240*x^9 + 3294720*x^7 - 532224*x^5 + 36960*x^3 - 720*x
3
16:
10027008*x^16 - 33423360*x^14 + 44728320*x^12 - 30750720*x^10 + 11531520*x^8 - 2306304*x^6 + 221760*x^4 - 7920*x^2 + 45
3
17:
22413312*x^17 - 80216064*x^15 + 116981760*x^13 - 89456640*x^11 + 38438400*x^9 - 9225216*x^7 + 1153152*x^5 - 63360*x^3 + 990*x
3
18:
49807360*x^18 - 190513152*x^16 + 300810240*x^14 - 253460480*x^12 + 123002880*x^10 - 34594560*x^8 + 5381376*x^6 - 411840*x^4 + 11880*x^2 - 55
3
19:
110100480*x^19 - 448266240*x^17 + 762052608*x^15 - 701890560*x^13 + 380190720*x^11 - 123002880*x^9 + 23063040*x^7 - 2306304*x^5 + 102960*x^3 - 1320*x
3
20:
242221056*x^20 - 1045954560*x^18 + 1905131520*x^16 - 1905131520*x^14 + 1140572160*x^12 - 418209792*x^10 + 92252160*x^8 - 11531520*x^6 + 720720*x^4 - 17160*x^2 + 66
7/2
0:
1
7/2
1:
7*x
7/2
2:
63/2*x^2 - 7/2
7/2
3:
231/2*x^3 - 63/2*x
7/2
4:
3003/8*x^4 - 693/4*x^2 + 63/8
7/2
5:
9009/8*x^5 - 3003/4*x^3 + 693/8*x
7/2
6:
51051/16*x^6 - 45045/16*x^4 + 9009/16*x^2 - 231/16
7/2
7:
138567/16*x^7 - 153153/16*x^5 + 45045/16*x^3 - 3003/16*x
7/2
8:
2909907/128*x^8 - 969969/32*x^6 + 765765/64*x^4 - 45045/32*x^2 + 3003/128
7/2
9:
7436429/128*x^9 - 2909907/32*x^7 + 2909907/64*x^5 - 255255/32*x^3 + 45045/128*x
7/2
10:
37182145/256*x^10 - 66927861/256*x^8 + 20369349/128*x^6 - 4849845/128*x^4 + 765765/256*x^2 - 9009/256
7/2
11:
91265265/256*x^11 - 185910725/256*x^9 + 66927861/128*x^7 - 20369349/128*x^5 + 4849845/256*x^3 - 153153/256*x
7/2
12:
882230895/1024*x^12 - 1003917915/512*x^10 + 1673196525/1024*x^8 - 156165009/256*x^6 + 101846745/1024*x^4 - 2909907/512*x^2 + 51051/1024
7/2
13:
2103781365/1024*x^13 - 2646692685/512*x^11 + 5019589575/1024*x^9 - 557732175/256*x^7 + 468495027/1024*x^5 - 20369349/512*x^3 + 969969/1024*x
7/2
14:
9917826435/2048*x^14 - 27349157745/2048*x^12 + 29113619535/2048*x^10 - 15058768725/2048*x^8 + 3904125225/2048*x^6 - 468495027/2048*x^4 + 20369349/2048*x^2 - 138567/2048
7/2
15:
23141595015/2048*x^15 - 69424785045/2048*x^13 + 82047473235/2048*x^11 - 48522699225/2048*x^9 + 15058768725/2048*x^7 - 2342475135/2048*x^5 + 156165009/2048*x^3 - 2909907/2048*x
7/2
16:
856239015555/32768*x^16 - 347123925225/4096*x^14 + 902522205585/8192*x^12 - 300840735195/4096*x^10 + 436704293025/16384*x^8 - 21082276215/4096*x^6 + 3904125225/8192*x^4 - 66927861/4096*x^2 + 2909907/32768
7/2
17:
1964313035685/32768*x^17 - 856239015555/4096*x^15 + 2429867476575/8192*x^13 - 902522205585/4096*x^11 + 1504203675975/16384*x^9 - 87340858605/4096*x^7 + 21082276215/8192*x^5 - 557732175/4096*x^3 + 66927861/32768*x
7/2
18:
8948537162565/65536*x^18 - 33393321606645/65536*x^16 + 12843585233325/16384*x^14 - 10529425731825/16384*x^12 + 9927744261435/32768*x^10 - 2707566616755/32768*x^8 + 203795336745/16384*x^6 - 15058768725/16384*x^4 + 1673196525/65536*x^2 - 7436429/65536
7/2
19:
20251952525805/65536*x^19 - 80536834463085/65536*x^17 + 33393321606645/16384*x^15 - 29968365544425/16384*x^13 + 31588277195475/32768*x^11 - 9927744261435/32768*x^9 + 902522205585/16384*x^7 - 87340858605/16384*x^5 + 15058768725/65536*x^3 - 185910725/65536*x
7/2
20:
182267572732245/262144*x^20 - 384787097990295/131072*x^18 + 1369126185872445/262144*x^16 - 166966608033225/32768*x^14 + 389588752077525/131072*x^12 - 69494209830045/65536*x^10 + 29783232784305/131072*x^8 - 902522205585/32768*x^6 + 436704293025/262144*x^4 - 5019589575/131072*x^2 + 37182145/262144
4
0:
1
4
1:
8*x
4
2:
40*x^2 - 4
4
3:
160*x^3 - 40*x
4
4:
560*x^4 - 240*x^2 + 10
4
5:
1792*x^5 - 1120*x^3 + 120*x
4
6:
5376*x^6 - 4480*x^4 + 840*x^2 - 20
4
7:
15360*x^7 - 16128*x^5 + 4480*x^3 - 280*x
4
8:
42240*x^8 - 53760*x^6 + 20160*x^4 - 2240*x^2 + 35
4
9:
112640*x^9 - 168960*x^7 + 80640*x^5 - 13440*x^3 + 560*x
4
10:
292864*x^10 - 506880*x^8 + 295680*x^6 - 67200*x^4 + 5040*x^2 - 56
4
11:
745472*x^11 - 1464320*x^9 + 1013760*x^7 - 295680*x^5 + 33600*x^3 - 1008*x
4
12:
1863680*x^12 - 4100096*x^10 + 3294720*x^8 - 1182720*x^6 + 184800*x^4 - 10080*x^2 + 84
4
13:
4587520*x^13 - 11182080*x^11 + 10250240*x^9 - 4392960*x^7 + 887040*x^5 - 73920*x^3 + 1680*x
4
14:
11141120*x^14 - 29818880*x^12 + 30750720*x^10 - 15375360*x^8 + 3843840*x^6 - 443520*x^4 + 18480*x^2 - 120
4
15:
26738688*x^15 - 77987840*x^13 + 89456640*x^11 - 51251200*x^9 + 15375360*x^7 - 2306304*x^5 + 147840*x^3 - 2640*x
4
16:
63504384*x^16 - 200540160*x^14 + 253460480*x^12 - 164003840*x^10 + 57657600*x^8 - 10762752*x^6 + 960960*x^4 - 31680*x^2 + 165
4
17:
149422080*x^17 - 508035072*x^15 + 701890560*x^13 - 506920960*x^11 + 205004800*x^9 - 46126080*x^7 + 5381376*x^5 - 274560*x^3 + 3960*x
4
18:
348651520*x^18 - 1270087680*x^16 + 1905131520*x^14 - 1520762880*x^12 + 697016320*x^10 - 184504320*x^8 + 26906880*x^6 - 1921920*x^4 + 51480*x^2 - 220
4
19:
807403520*x^19 - 3137863680*x^17 + 5080350720*x^15 - 4445306880*x^13 + 2281144320*x^11 - 697016320*x^9 + 123002880*x^7 - 11531520*x^5 + 480480*x^3 - 5720*x
4
20:
1857028096*x^20 - 7670333440*x^18 + 13335920640*x^16 - 12700876800*x^14 + 7223623680*x^12 - 2509258752*x^10 + 522762240*x^8 - 61501440*x^6 + 3603600*x^4 - 80080*x^2 + 286
Definition
The Gegenbauer polynomials $C_n^{\alpha}$, $n\geq 0$, can be defined via the recurrence formula (1).
Parameters
$\alpha$
—   real number
$n$
—   integer ($n \geq 0$)
Formulas
(1)
$C_0^{\alpha}(x) = 1$, $C_1^{\alpha}(x) = 2\alpha x$, and $(n+1)C_{n+1}^{\alpha}(x) =
2x(n+\alpha) C_n^{\alpha}(x) - (n+2\alpha-1) C_{n-1}^{\alpha}(x)$
for $n\geq 1$ (recurrence formula).
(2)
$\sum_{n=0}^\infty C_n^{\alpha}(x)t^n = \frac{1}{(1-2tx+t^2)^\alpha}$ (generating function).
Comments
(3)
The $C_n^{\alpha}$ are orthogonal with respect to the inner product $\langle f,g\rangle = \int_{-1}^1 f(x)g(x) (1-x^2)^{\alpha-1/2} dx$.
Programs
(P1)
Sage
polynomials = {alpha: 
  {n: gegenbauer(n,alpha,x) for n in [0..10]}
  for alpha in [-1/2,0,1/2,1]
}
Links
Data properties
Entries are of type: rational polynomial
Table is complete: no