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+Title: Gegenbauer polynomials+Definition: The Gegenbauer polynomials $C_n^{\alpha}$, $n\geq 0$, can be defined+ via the recurrence formula CITE{formula-recurrence}.+Parameters:+ alpha:+ display: $\alpha$+ type: R+ n:+ type: Z+ constraints: $n \geq 0$+Comments:+ comment-orthogonal: The $C_n^{\alpha}$ are orthogonal with respect to the inner+ product $\langle f,g\rangle = \int_{-1}^1 f(x)g(x) (1-x^2)^{\alpha-1/2} dx$.+Formulas:+ formula-recurrence: "$C_0^{\\alpha}(x) = 1$, $C_1^{\\alpha}(x) = 2\\alpha x$, \+ \ and $(n+1)C_{n+1}^{\\alpha}(x) = \n 2x(n+\\alpha) C_n^{\\alpha}(x) - (n+2\\\+ alpha-1) C_{n-1}^{\\alpha}(x)$ \nfor $n\\geq 1$ (recurrence formula)."+ forumla-generating-function: $\sum_{n=0}^\infty C_n^{\alpha}(x)t^n = \frac{1}{(1-2tx+t^2)^\alpha}$ (generating+ function).+Programs:+ program-sage:+ language: Sage+ code: "polynomials = {alpha: \n {n: gegenbauer(n,alpha,x) for n in [0..10]}\n\+ \ for alpha in [-1/2,0,1/2,1]\n}"+References: {}+Links:+ MathWorld:+ title: 'MathWorld: Gegenbauer polynomials'+ url: https://mathworld.wolfram.com/GegenbauerPolynomial.html+ Wiki:+ title: 'Wikipedia: Gegenbauer polynomials'+ url: https://en.wikipedia.org/wiki/Gegenbauer_polynomials+Similar tables: ''+Keywords: ''+Tags:+- polynomial+- orthogonal polynomials+Data properties:+ type: Q[]+ complete: 'no'+Display properties:+ number-header: $C_n^{\alpha}(x)$+Data:+ a:+ '0': '1'+ '1': 2*a*x+ '2': 2*(a + 1)*a*x^2 - a+ '3': 4/3*(a + 2)*(a + 1)*a*x^3 - 2*(a + 1)*a*x+ '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+ '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+ '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+ '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+ '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+ '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+ '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+ '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+ '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+ '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+ '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+ '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': -2*x+ '2': '1'+ '3': '0'+ -3/4:+ '0': '1'+ '1': -3/2*x+ '2': -3/8*x^2 + 3/4+ '3': -5/16*x^3 + 3/8*x+ '4': -45/128*x^4 + 15/32*x^2 - 3/32+ '5': -117/256*x^5 + 45/64*x^3 - 15/64*x+ '6': -663/1024*x^6 + 585/512*x^4 - 135/256*x^2 + 5/128+ '7': -1989/2048*x^7 + 1989/1024*x^5 - 585/512*x^3 + 45/256*x+ '8': -49725/32768*x^8 + 13923/4096*x^6 - 9945/4096*x^4 + 585/1024*x^2 - 45/2048+ '9': -160225/65536*x^9 + 49725/8192*x^7 - 41769/8192*x^5 + 3315/2048*x^3 - 585/4096*x+ '10': -1057485/262144*x^10 + 1442025/131072*x^8 - 348075/32768*x^6 + 69615/16384*x^4+ - 9945/16384*x^2 + 117/8192+ '11': -3556995/524288*x^11 + 5287425/262144*x^9 - 1442025/65536*x^7 + 348075/32768*x^5+ - 69615/32768*x^3 + 1989/16384*x+ '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+ '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+ '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+ '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+ '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+ '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+ '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+ '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+ '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'+ '1': -4/3*x+ '2': -4/9*x^2 + 2/3+ '3': -32/81*x^3 + 4/9*x+ '4': -112/243*x^4 + 16/27*x^2 - 1/9+ '5': -448/729*x^5 + 224/243*x^3 - 8/27*x+ '6': -5824/6561*x^6 + 1120/729*x^4 - 56/81*x^2 + 4/81+ '7': -26624/19683*x^7 + 5824/2187*x^5 - 1120/729*x^3 + 56/243*x+ '8': -126464/59049*x^8 + 93184/19683*x^6 - 7280/2187*x^4 + 560/729*x^2 - 7/243+ '9': -5564416/1594323*x^9 + 505856/59049*x^7 - 46592/6561*x^5 + 14560/6561*x^3+ - 140/729*x+ '10': -27822080/4782969*x^10 + 2782208/177147*x^8 - 885248/59049*x^6 + 116480/19683*x^4+ - 1820/2187*x^2 + 14/729+ '11': -141639680/14348907*x^11 + 139110400/4782969*x^9 - 5564416/177147*x^7 ++ 885248/59049*x^5 - 58240/19683*x^3 + 364/2187*x+ '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+ '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+ '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+ '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+ '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+ '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+ '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+ '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+ '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': -x+ '2': -1/2*x^2 + 1/2+ '3': -1/2*x^3 + 1/2*x+ '4': -5/8*x^4 + 3/4*x^2 - 1/8+ '5': -7/8*x^5 + 5/4*x^3 - 3/8*x+ '6': -21/16*x^6 + 35/16*x^4 - 15/16*x^2 + 1/16+ '7': -33/16*x^7 + 63/16*x^5 - 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308/729+ '11': 9660465152/14348907*x^11 - 8301962240/4782969*x^9 + 286274560/177147*x^7+ - 38536960/59049*x^5 + 2094400/19683*x^3 - 10472/2187*x+ '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+ '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+ '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+ '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+ '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+ '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+ '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+ '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+ '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'+ '1': 3/2*x+ '2': 21/8*x^2 - 3/4+ '3': 77/16*x^3 - 21/8*x+ '4': 1155/128*x^4 - 231/32*x^2 + 21/32+ '5': 4389/256*x^5 - 1155/64*x^3 + 231/64*x+ '6': 33649/1024*x^6 - 21945/512*x^4 + 3465/256*x^2 - 77/128+ '7': 129789/2048*x^7 - 100947/1024*x^5 + 21945/512*x^3 - 1155/256*x+ '8': 4023459/32768*x^8 - 908523/4096*x^6 + 504735/4096*x^4 - 21945/1024*x^2 ++ 1155/2048+ '9': 15646785/65536*x^9 - 4023459/8192*x^7 + 2725569/8192*x^5 - 168245/2048*x^3+ + 21945/4096*x+ '10': 122044923/262144*x^10 - 140821065/131072*x^8 + 28164213/32768*x^6 - 4542615/16384*x^4+ + 504735/16384*x^2 - 4389/8192+ '11': 477084699/524288*x^11 - 610224615/262144*x^9 + 140821065/65536*x^7 - 28164213/32768*x^5+ + 4542615/32768*x^3 - 100947/16384*x+ '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+ '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+ '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+ '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+ '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+ '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+ '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+ '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+ '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+ '1':+ equals: HREF{Chebyshev_polynomials_of_the_second_kind}+ 3/2:+ '0': '1'+ '1': 3*x+ '2': 15/2*x^2 - 3/2+ '3': 35/2*x^3 - 15/2*x+ '4': 315/8*x^4 - 105/4*x^2 + 15/8+ '5': 693/8*x^5 - 315/4*x^3 + 105/8*x+ '6': 3003/16*x^6 - 3465/16*x^4 + 945/16*x^2 - 35/16+ '7': 6435/16*x^7 - 9009/16*x^5 + 3465/16*x^3 - 315/16*x+ '8': 109395/128*x^8 - 45045/32*x^6 + 45045/64*x^4 - 3465/32*x^2 + 315/128+ '9': 230945/128*x^9 - 109395/32*x^7 + 135135/64*x^5 - 15015/32*x^3 + 3465/128*x+ '10': 969969/256*x^10 - 2078505/256*x^8 + 765765/128*x^6 - 225225/128*x^4 + 45045/256*x^2+ - 693/256+ '11': 2028117/256*x^11 - 4849845/256*x^9 + 2078505/128*x^7 - 765765/128*x^5 ++ 225225/256*x^3 - 9009/256*x+ '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+ '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+ '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+ '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+ '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+ '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+ '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+ '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+ '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'+ '1': 4*x+ '2': 12*x^2 - 2+ '3': 32*x^3 - 12*x+ '4': 80*x^4 - 48*x^2 + 3+ '5': 192*x^5 - 160*x^3 + 24*x+ '6': 448*x^6 - 480*x^4 + 120*x^2 - 4+ '7': 1024*x^7 - 1344*x^5 + 480*x^3 - 40*x+ '8': 2304*x^8 - 3584*x^6 + 1680*x^4 - 240*x^2 + 5+ '9': 5120*x^9 - 9216*x^7 + 5376*x^5 - 1120*x^3 + 60*x+ '10': 11264*x^10 - 23040*x^8 + 16128*x^6 - 4480*x^4 + 420*x^2 - 6+ '11': 24576*x^11 - 56320*x^9 + 46080*x^7 - 16128*x^5 + 2240*x^3 - 84*x+ '12': 53248*x^12 - 135168*x^10 + 126720*x^8 - 53760*x^6 + 10080*x^4 - 672*x^2+ + 7+ '13': 114688*x^13 - 319488*x^11 + 337920*x^9 - 168960*x^7 + 40320*x^5 - 4032*x^3+ + 112*x+ '14': 245760*x^14 - 745472*x^12 + 878592*x^10 - 506880*x^8 + 147840*x^6 - 20160*x^4+ + 1008*x^2 - 8+ '15': 524288*x^15 - 1720320*x^13 + 2236416*x^11 - 1464320*x^9 + 506880*x^7 - 88704*x^5+ + 6720*x^3 - 144*x+ '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+ '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+ '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+ '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+ '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'+ '1': 5*x+ '2': 35/2*x^2 - 5/2+ '3': 105/2*x^3 - 35/2*x+ '4': 1155/8*x^4 - 315/4*x^2 + 35/8+ '5': 3003/8*x^5 - 1155/4*x^3 + 315/8*x+ '6': 15015/16*x^6 - 15015/16*x^4 + 3465/16*x^2 - 105/16+ '7': 36465/16*x^7 - 45045/16*x^5 + 15015/16*x^3 - 1155/16*x+ '8': 692835/128*x^8 - 255255/32*x^6 + 225225/64*x^4 - 15015/32*x^2 + 1155/128+ '9': 1616615/128*x^9 - 692835/32*x^7 + 765765/64*x^5 - 75075/32*x^3 + 15015/128*x+ '10': 7436429/256*x^10 - 14549535/256*x^8 + 4849845/128*x^6 - 1276275/128*x^4+ + 225225/256*x^2 - 3003/256+ '11': 16900975/256*x^11 - 37182145/256*x^9 + 14549535/128*x^7 - 4849845/128*x^5+ + 1276275/256*x^3 - 45045/256*x+ '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+ '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+ '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+ '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+ '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+ '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+ '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+ '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+ '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'+ '1': 6*x+ '2': 24*x^2 - 3+ '3': 80*x^3 - 24*x+ '4': 240*x^4 - 120*x^2 + 6+ '5': 672*x^5 - 480*x^3 + 60*x+ '6': 1792*x^6 - 1680*x^4 + 360*x^2 - 10+ '7': 4608*x^7 - 5376*x^5 + 1680*x^3 - 120*x+ '8': 11520*x^8 - 16128*x^6 + 6720*x^4 - 840*x^2 + 15+ '9': 28160*x^9 - 46080*x^7 + 24192*x^5 - 4480*x^3 + 210*x+ '10': 67584*x^10 - 126720*x^8 + 80640*x^6 - 20160*x^4 + 1680*x^2 - 21+ '11': 159744*x^11 - 337920*x^9 + 253440*x^7 - 80640*x^5 + 10080*x^3 - 336*x+ '12': 372736*x^12 - 878592*x^10 + 760320*x^8 - 295680*x^6 + 50400*x^4 - 3024*x^2+ + 28+ '13': 860160*x^13 - 2236416*x^11 + 2196480*x^9 - 1013760*x^7 + 221760*x^5 - 20160*x^3+ + 504*x+ '14': 1966080*x^14 - 5591040*x^12 + 6150144*x^10 - 3294720*x^8 + 887040*x^6 -+ 110880*x^4 + 5040*x^2 - 36+ '15': 4456448*x^15 - 13762560*x^13 + 16773120*x^11 - 10250240*x^9 + 3294720*x^7+ - 532224*x^5 + 36960*x^3 - 720*x+ '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+ '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+ '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+ '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+ '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'+ '1': 7*x+ '2': 63/2*x^2 - 7/2+ '3': 231/2*x^3 - 63/2*x+ '4': 3003/8*x^4 - 693/4*x^2 + 63/8+ '5': 9009/8*x^5 - 3003/4*x^3 + 693/8*x+ '6': 51051/16*x^6 - 45045/16*x^4 + 9009/16*x^2 - 231/16+ '7': 138567/16*x^7 - 153153/16*x^5 + 45045/16*x^3 - 3003/16*x+ '8': 2909907/128*x^8 - 969969/32*x^6 + 765765/64*x^4 - 45045/32*x^2 + 3003/128+ '9': 7436429/128*x^9 - 2909907/32*x^7 + 2909907/64*x^5 - 255255/32*x^3 + 45045/128*x+ '10': 37182145/256*x^10 - 66927861/256*x^8 + 20369349/128*x^6 - 4849845/128*x^4+ + 765765/256*x^2 - 9009/256+ '11': 91265265/256*x^11 - 185910725/256*x^9 + 66927861/128*x^7 - 20369349/128*x^5+ + 4849845/256*x^3 - 153153/256*x+ '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+ '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+ '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+ '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+ '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+ '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+ '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+ '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+ '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'+ '1': 8*x+ '2': 40*x^2 - 4+ '3': 160*x^3 - 40*x+ '4': 560*x^4 - 240*x^2 + 10+ '5': 1792*x^5 - 1120*x^3 + 120*x+ '6': 5376*x^6 - 4480*x^4 + 840*x^2 - 20+ '7': 15360*x^7 - 16128*x^5 + 4480*x^3 - 280*x+ '8': 42240*x^8 - 53760*x^6 + 20160*x^4 - 2240*x^2 + 35+ '9': 112640*x^9 - 168960*x^7 + 80640*x^5 - 13440*x^3 + 560*x+ '10': 292864*x^10 - 506880*x^8 + 295680*x^6 - 67200*x^4 + 5040*x^2 - 56+ '11': 745472*x^11 - 1464320*x^9 + 1013760*x^7 - 295680*x^5 + 33600*x^3 - 1008*x+ '12': 1863680*x^12 - 4100096*x^10 + 3294720*x^8 - 1182720*x^6 + 184800*x^4 - 10080*x^2+ + 84+ '13': 4587520*x^13 - 11182080*x^11 + 10250240*x^9 - 4392960*x^7 + 887040*x^5 -+ 73920*x^3 + 1680*x+ '14': 11141120*x^14 - 29818880*x^12 + 30750720*x^10 - 15375360*x^8 + 3843840*x^6+ - 443520*x^4 + 18480*x^2 - 120+ '15': 26738688*x^15 - 77987840*x^13 + 89456640*x^11 - 51251200*x^9 + 15375360*x^7+ - 2306304*x^5 + 147840*x^3 - 2640*x+ '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+ '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+ '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+ '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+ '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
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