チェビシェフ多項式の具体的表示

ルジャンドル多項式の次にチェビジェフ多項式が出てきた。
これは、

$\Large T_{n}(cos~\theta)=cos~n\theta$

と表せるもので、
例えば、

$\large cos~2\theta=2cos^2\theta-1$

だから、

$\large T_{2}(x)=2x^2-1$

cosnθの計算って高校のとき誰もがやったと思う。
なかなか計算が大変で、5次くらいまでしかやった記憶がない。
ここではなるべく高次まで計算したいと思う。

$\large cos(n+1)\theta=cos~n\theta~cos~\theta-sin~n\theta~sin~\theta$
$\large sin(n+1)\theta=sin~n\theta~cos~\theta+cos~n\theta~sin~\theta$

上を下に代入してsin(n+1)θ、sinnθを消去すると、

$\large cos(n+2)\theta=2cos~\theta~cos(n+1)\theta-cosn~\theta$

となって、

$\Large T_{n+2}(x)=2xT_{n+1}(x)-T_{n}(x)$

という漸化式が得られる。
これを用いてプログラムを書くと、63次まで得られた。

T₀(x)=1
T₁(x)=x
T₂(x)=-1+2x²
T₃(x)=-3x+4x³
T₄(x)=1-8x²+8x⁴
T₅(x)=5x-20x³+16x⁵
T₆(x)=-1+18x²-48x⁴+32x⁶
T₇(x)=-7x+56x³-112x⁵+64x⁷
T₈(x)=1-32x²+160x⁴-256x⁶+128x⁸
T₉(x)=9x-120x³+432x⁵-576x⁷+256x⁹
T₁₀(x)=-1+50x²-400x⁴+1120x⁶-1280x⁸+512x¹⁰
T₁₁(x)=-11x+220x³-1232x⁵+2816x⁷-2816x⁹+1024x¹¹
T₁₂(x)=1-72x²+840x⁴-3584x⁶+6912x⁸-6144x¹⁰+2048x¹²
T₁₃(x)=13x-364x³+2912x⁵-9984x⁷+16640x⁹-13312x¹¹+4096x¹³
T₁₄(x)=-1+98x²-1568x⁴+9408x⁶-26880x⁸+39424x¹⁰-28672x¹²+8192x¹⁴
T₁₅(x)=-15x+560x³-6048x⁵+28800x⁷-70400x⁹+92160x¹¹-61440x¹³+16384x¹⁵
T₁₆(x)=1-128x²+2688x⁴-21504x⁶+84480x⁸-180224x¹⁰+212992x¹²-131072x¹⁴+32768x¹⁶
T₁₇(x)=17x-816x³+11424x⁵-71808x⁷+239360x⁹-452608x¹¹+487424x¹³-278528x¹⁵+65536x¹⁷
T₁₈(x)=-1+162x²-4320x⁴+44352x⁶-228096x⁸+658944x¹⁰-1118208x¹²+1105920x¹⁴-589824x¹⁶+131072x¹⁸
T₁₉(x)=-19x+1140x³-20064x⁵+160512x⁷-695552x⁹+1770496x¹¹-2723840x¹³+2490368x¹⁵-1245184x¹⁷+262144x¹⁹
T₂₀(x)=1-200x²+6600x⁴-84480x⁶+549120x⁸-2050048x¹⁰+4659200x¹²-6553600x¹⁴+5570560x¹⁶-2621440x¹⁸+524288x²⁰
T₂₁(x)=21x-1540x³+33264x⁵-329472x⁷+1793792x⁹-5870592x¹¹+12042240x¹³-15597568x¹⁵+12386304x¹⁷-5505024x¹⁹+1048576x²¹
T₂₂(x)=-1+242x²-9680x⁴+151008x⁶-1208064x⁸+5637632x¹⁰-16400384x¹²+30638080x¹⁴-36765696x¹⁶+27394048x¹⁸-11534336x²⁰+2097152x²²
T₂₃(x)=-23x+2024x³-52624x⁵+631488x⁷-4209920x⁹+17145856x¹¹-44843008x¹³+76873728x¹⁵-85917696x¹⁷+60293120x¹⁹-24117248x²¹+4194304x²³
T₂₄(x)=1-288x²+13728x⁴-256256x⁶+2471040x⁸-14057472x¹⁰+50692096x¹²-120324096x¹⁴+190513152x¹⁶-199229440x¹⁸+132120576x²⁰-50331648x²²+8388608x²⁴
T₂₅(x)=25x-2600x³+80080x⁵-1144000x⁷+9152000x⁹-45260800x¹¹+146227200x¹³-317521920x¹⁵+466944000x¹⁷-458752000x¹⁹+288358400x²¹-104857600x²³+16777216x²⁵
T₂₆(x)=-1+338x²-18928x⁴+416416x⁶-4759040x⁸+32361472x¹⁰-141213696x¹²+412778496x¹⁴-825556992x¹⁶+1133117440x¹⁸-1049624576x²⁰+627048448x²²-218103808x²⁴+33554432x²⁶
T₂₇(x)=-27x+3276x³-117936x⁵+1976832x⁷-18670080x⁹+109983744x¹¹-428654592x¹³+1143078912x¹⁵-2118057984x¹⁷+2724986880x¹⁹-2387607552x²¹+1358954496x²³-452984832x²⁵+67108864x²⁷
T₂₈(x)=1-392x²+25480x⁴-652288x⁶+8712704x⁸-69701632x¹⁰+361181184x¹²-1270087680x¹⁴+3111714816x¹⁶-5369233408x¹⁸+6499598336x²⁰-5402263552x²²+2936012800x²⁴-939524096x²⁶+134217728x²⁸
T₂₉(x)=29x-4060x³+168896x⁵-3281408x⁷+36095488x⁹-249387008x¹¹+1151016960x¹³-3683254272x¹⁵+8341487616x¹⁷-13463453696x¹⁹+15386804224x²¹-12163481600x²³+6325010432x²⁵-1946157056x²⁷+268435456x²⁹
T₃₀(x)=-1+450x²-33600x⁴+990080x⁶-15275520x⁸+141892608x¹⁰-859955200x¹²+3572121600x¹⁴-10478223360x¹⁶+22052208640x¹⁸-33426505728x²⁰+36175872000x²²-27262976000x²⁴+13589544960x²⁶-4026531840x²⁸+536870912x³⁰
T₃₁(x)=-31x+4960x³-236096x⁵+5261568x⁷-66646528x⁹+533172224x¹¹-2870927360x¹³+10827497472x¹⁵-29297934336x¹⁷+57567870976x¹⁹-82239815680x²¹+84515225600x²³-60850962432x²⁵+29125246976x²⁷-8321499136x²⁹+1073741824x³¹
T₃₂(x)=1-512x²+43520x⁴-1462272x⁶+25798656x⁸-275185664x¹⁰+1926299648x¹²-9313976320x¹⁴+32133218304x¹⁶-80648077312x¹⁸+148562247680x²⁰-200655503360x²²+196293427200x²⁴-135291469824x²⁶+62277025792x²⁸-17179869184x³⁰+2147483648x³²
T₃₃(x)=33x-5984x³+323136x⁵-8186112x⁷+118243840x⁹-1083543552x¹¹+6723526656x¹³-29455450112x¹⁵+93564370944x¹⁷-218864025600x¹⁹+379364311040x²¹-485826232320x²³+453437816832x²⁵-299708186624x²⁷+132875550720x²⁹-35433480192x³¹+4294967296x³³
T₃₄(x)=-1+578x²-55488x⁴+2108544x⁶-42170880x⁸+511673344x¹⁰-4093386752x¹²+22761029632x¹⁴-91044118528x¹⁶+267776819200x¹⁸-586290298880x²⁰+959384125440x²²-1167945891840x²⁴+1042167103488x²⁶-661693399040x²⁸+282930970624x³⁰-73014444032x³²+8589934592x³⁴
T₃₅(x)=-35x+7140x³-434112x⁵+12403200x⁷-202585600x⁹+2106890240x¹¹-14910300160x¹³+74977509376x¹⁵-275652608000x¹⁷+754417664000x¹⁹-1551944908800x²¹+2404594483200x²³-2789329600512x²⁵+2384042393600x²⁷-1456262348800x²⁹+601295421440x³¹-150323855360x³³+17179869184x³⁵
T₃₆(x)=1-648x²+69768x⁴-2976768x⁶+66977280x⁸-916844544x¹⁰+8307167232x¹²-52581629952x¹⁴+240999137280x¹⁶-819082035200x¹⁸+2095125626880x²⁰-4063273943040x²²+5977134858240x²⁴-6620826304512x²⁶+5429778186240x²⁸-3195455668224x³⁰+1275605286912x³²-309237645312x³⁴+34359738368x³⁶
T₃₇(x)=37x-8436x³+573648x⁵-18356736x⁷+336540160x⁹-3940579328x¹¹+31524634624x¹³-180140769280x¹⁵+757650882560x¹⁷-2392581734400x¹⁹+5742196162560x²¹-10531142369280x²³+14743599316992x²⁵-15625695002624x²⁷+12315818721280x²⁹-6992206757888x³¹+2701534429184x³³-635655159808x³⁵+68719476736x³⁷
T₃₈(x)=-1+722x²-86640x⁴+4124064x⁶-103690752x⁸+1589924864x¹⁰-16188325888x¹²+115630899200x¹⁴-601280675840x¹⁶+2334383800320x¹⁸-6880289095680x²⁰+15547666268160x²²-27039419596800x²⁴+36108024938496x²⁶-36681168191488x²⁸+27827093110784x³⁰-15260018802688x³²+5712306503680x³⁴-1305670057984x³⁶+137438953472x³⁸
T₃₉(x)=-39x+9880x³-746928x⁵+26604864x⁷-543921664x⁹+7120429056x¹¹-63901286400x¹³+411402567680x¹⁵-1960212234240x¹⁷+7061349335040x¹⁹-19502774353920x²¹+41626474905600x²³-68822438510592x²⁵+87841744879616x²⁷-85678155104256x²⁹+62646392979456x³¹-33221572034560x³³+12060268167168x³⁵-2680059592704x³⁷+274877906944x³⁹
T₄₀(x)=1-800x²+106400x⁴-5617920x⁶+156900480x⁸-2677768192x¹⁰+30429184000x¹²-243433472000x¹⁴+1424085811200x¹⁶-6254808268800x¹⁸+21002987765760x²⁰-54553214976000x²²+110292369408000x²⁴-173752901959680x²⁶+212364657950720x²⁸-199183403319296x³⁰+140552804761600x³²-72155450572800x³⁴+25426206392320x³⁶-5497558138880x³⁸+549755813888x⁴⁰
T₄₁(x)=41x-11480x³+959728x⁵-37840704x⁷+857722624x⁹-12475965440x¹¹+124759654400x¹³-898269511680x¹⁵+4808383856640x¹⁷-19570965872640x¹⁹+61508749885440x²¹-150732904857600x²³+289407177326592x²⁵-435347548798976x²⁷+510407471005696x²⁹-461013199618048x³¹+314327181557760x³³-156371169312768x³⁵+53532472377344x³⁷-11269994184704x³⁹+1099511627776x⁴¹
T₄₂(x)=-1+882x²-129360x⁴+7537376x⁶-232581888x⁸+4393213440x¹⁰-55381114880x¹²+492952780800x¹⁴-3220624834560x¹⁶+15871575982080x¹⁸-60144919511040x²⁰+177570714746880x²²-411758179123200x²⁴+752567256612864x²⁶-1083059755548672x²⁸+1219998345330688x³⁰-1062579203997696x³²+700809813688320x³⁴-338168545017856x³⁶+112562502893568x³⁸-23089744183296x⁴⁰+2199023255552x⁴²
T₄₃(x)=-43x+13244x³-1218448x⁵+52915456x⁷-1322886400x⁹+21262392320x¹¹-235521884160x¹³+1884175073280x¹⁵-11249633525760x¹⁷+51314117836800x¹⁹-181798588907520x²¹+505874334351360x²³-1112923535572992x²⁵+1940482062024704x²⁷-2676526982103040x²⁹+2901009890279424x³¹-2439485589553152x³³+1557990796689408x³⁵-729869562413056x³⁷+236394999971840x³⁹-47278999994368x⁴¹+4398046511104x⁴³
T₄₄(x)=1-968x²+155848x⁴-9974272x⁶+338412800x⁸-7038986240x¹⁰+97905899520x¹²-963996549120x¹⁴+6988974981120x¹⁶-38370843033600x¹⁸+162773155184640x²⁰-541167892561920x²²+1423506847825920x²⁴-2978414327758848x²⁶+4964023879598080x²⁸-6573052309536768x³⁰+6864598984556544x³²-5579780992794624x³⁴+3454150138396672x³⁶-1572301627719680x³⁸+495879744126976x⁴⁰-96757023244288x⁴²+8796093022208x⁴⁴
T₄₅(x)=45x-15180x³+1530144x⁵-72864000x⁷+1999712000x⁹-35340364800x¹¹+431333683200x¹³-3812168171520x¹⁵+25227583488000x¹⁷-128055803904000x¹⁹+507344899276800x²¹-1588210119475200x²³+3959937231224832x²⁵-7897310717542400x²⁷+12604574741299200x²⁹-16047114509352960x³¹+16168683558666240x³³-12717552782278656x³⁵+7638169839206400x³⁷-3380998255411200x³⁹+1039038488248320x⁴¹-197912092999680x⁴³+17592186044416x⁴⁵
T₄₆(x)=-1+1058x²-186208x⁴+13034560x⁶-484140800x⁸+11038410240x¹⁰-168586629120x¹²+1826663915520x¹⁴-14613311324160x¹⁶+88826010009600x¹⁸-418884762992640x²⁰+1555857691115520x²²-4599927086776320x²⁴+10898288790208512x²⁶-20758645314682880x²⁸+31782201792135168x³⁰-38958828003262464x³²+37917148110127104x³⁴-28889255702953984x³⁶+16848641306132480x³⁸-7257876254949376x⁴⁰+2174833999740928x⁴²-404620279021568x⁴⁴+35184372088832x⁴⁶
T₄₇(x)=-47x+17296x³-1902560x⁵+98933120x⁷-2967993600x⁹+57417185280x¹¹-768506941440x¹³+7465496002560x¹⁵-54454206136320x¹⁷+305707823923200x¹⁹-1345114425262080x²¹+4699925501706240x²³-13159791404777472x²⁵+29693888297959424x²⁷-54121865370664960x²⁹+79611518093623296x³¹-94086339565191168x³³+88551849002532864x³⁵-65416681245114368x³⁷+37078280867676160x³⁹-15554790998147072x⁴¹+4547580092481536x⁴³-826832744087552x⁴⁵+70368744177664x⁴⁷
T₄₈(x)=1-1152x²+220800x⁴-16839680x⁶+682007040x⁸-16974397440x¹⁰+283420999680x¹²-3363677798400x¹⁴+29544303329280x¹⁶-197734422282240x¹⁸+1030300410839040x²⁰-4246086541639680x²²+13999778090188800x²⁴-37217871599763456x²⁶+80146421910601728x²⁸-140025932533465088x³⁰+198181864190509056x³²-226089827240509440x³⁴+205992953708019712x³⁶-147682003796361216x³⁸+81414437990301696x⁴⁰-33284415996035072x⁴²+9499780463984640x⁴⁴-1688849860263936x⁴⁶+140737488355328x⁴⁸
T₄₉(x)=49x-19600x³+2344160x⁵-132612480x⁷+4332007680x⁹-91365980160x¹¹+1335348940800x¹³-14192851599360x¹⁵+113542812794880x¹⁷-701176668487680x¹⁹+3405715246940160x²¹-13192098584985600x²³+41159347585155072x²⁵-104129631497486336x²⁷+214414709191868416x²⁹-359663383160553472x³¹+490450067946209280x³³-540731503483551744x³⁵+477402588661153792x³⁷-332442288460398592x³⁹+178383666978750464x⁴¹-71116412084551680x⁴³+19826393672056832x⁴⁵-3448068464705536x⁴⁷+281474976710656x⁴⁹
T₅₀(x)=-1+1250x²-260000x⁴+21528000x⁶-947232000x⁸+25638412800x¹⁰-466152960000x¹²+6034375680000x¹⁴-57930006528000x¹⁶+424820047872000x¹⁸-2432653747814400x²⁰+11057517035520000x²²-40383975260160000x²⁴+119536566770073600x²⁶-288405684905574400x²⁸+568855350917201920x³⁰-917508630511616000x³²+1206989963132928000x³⁴-1287455960675123200x³⁶+1102487181118668800x³⁸-746299014911098880x⁴⁰+390051749953536000x⁴²-151732604633088000x⁴⁴+41341637204377600x⁴⁶-7036874417766400x⁴⁸+562949953421312x⁵⁰
T₅₁(x)=-51x+22100x³-2864160x⁵+175668480x⁷-6226471680x⁹+142642805760x¹¹-2267654860800x¹³+26261602959360x¹⁵-229402825850880x¹⁷+1550816764231680x¹⁹-8271022742568960x²¹+35307132656025600x²³-121927298105475072x²⁵+343202765037633536x²⁷-791226079003017216x²⁹+1497374084994957312x³¹-2325467328969441280x³³+2954711429749407744x³⁵-3052314510011400192x³⁷+2537416650697736192x³⁹-1670981696800948224x⁴¹+851219911991623680x⁴³-323291602938232832x⁴⁵+86131342873460736x⁴⁷-14355223812243456x⁴⁹+1125899906842624x⁵¹
T₅₂(x)=1-1352x²+304200x⁴-27256320x⁶+1298568960x⁸-38091356160x¹⁰+751438571520x¹²-10569685401600x¹⁴+110453212446720x¹⁶-883625699573760x¹⁸+5534287276277760x²⁰-27599562520657920x²²+110998240572211200x²⁴-363391162981023744x²⁶+974811214980841472x²⁸-2151307508923236352x³⁰+3912256800501530624x³²-5857924621071810560x³⁴+7196878820173938688x³⁶-7207116201141469184x³⁸+5821132316306571264x⁴⁰-3732015143555432448x⁴²+1854172428616335360x⁴⁴-687924843080843264x⁴⁶+179299560164687872x⁴⁸-29273397577908224x⁵⁰+2251799813685248x⁵²
T₅₃(x)=53x-24804x³+3472560x⁵-230181120x⁷+8823609600x⁹-218825518080x¹¹+3770532003840x¹³-47400973762560x¹⁵+450309250744320x¹⁷-3318068163379200x¹⁹+19339597295124480x²¹-90506257697341440x²³+343923779249897472x²⁵-1069985090999681024x²⁷+2740848508964700160x²⁹-5799989102841430016x³¹-8296763143737049088x³³+3776183401816522752x³⁵-1000671923350274048x³⁷+1495095020728877056x³⁹-5133497744295460864x⁴¹-8315250199102488576x⁴³+4031636460170903552x⁴⁵-1461981029035147264x⁴⁷+372954344141619200x⁴⁹-59672695062659072x⁵¹+4503599627370496x⁵³
T₅₄(x)=-1+1458x²-353808x⁴+34201440x⁶-1758931200x⁸+55738575360x¹⁰-1189089607680x¹²+18110749409280x¹⁴-205255159971840x¹⁶+1784244201062400x¹⁸-12170423603036160x²⁰+66278757110906880x²²-292010755966894080x²⁴+1051238721480818688x²⁶-3114781396980203520x²⁸+7633004526852636672x³⁰+2934509067525160960x³²+7711142407307264000x³⁴+355487983459106816x³⁶+5205772354440921088x³⁸-2830942274848817152x⁴⁰-6534980345035489280x⁴²-37928753111760896x⁴⁴+8751197763422650368x⁴⁶-3103261618234982400x⁴⁸+775182085861146624x⁵⁰-121597189939003392x⁵²+9007199254740992x⁵⁴
T₅₅(x)=-55x+27720x³-4180176x⁵+298584000x⁷-12341472000x⁹+330302668800x¹¹-6148711219200x¹³+83622472581120x¹⁵-860819570688000x¹⁷+6886556565504000x¹⁹-43680444501196800x²¹+223063771919155200x²³-927945291183685632x²⁵+3172462533961318400x²⁷-8970411302925107200x²⁹+2619254082837151744x³¹-4280962794922180608x³³-6800642660911546368x³⁵+1711647890268487680x³⁷+8916449688152965120x³⁹-528386805402173440x⁴¹-4754710490968489984x⁴³-4107493966394425344x⁴⁵+517632482170896384x⁴⁷-6579477580611584000x⁴⁹+1610036866784952320x⁵¹-247697979505377280x⁵³+18014398509481984x⁵⁵
T₅₆(x)=1-1568x²+409248x⁴-42561792x⁶+2356099200x⁸-80421519360x¹⁰+1849694945280x¹²-30408171847680x¹⁴+372500105134080x¹⁶-3505883342438400x¹⁸+25943536734044160x²⁰-153639646113300480x²²+738138299805204480x²⁴-2907129303848189952x²⁶-8987037608806711296x²⁸-7127083058993299456x³⁰+2303999098149142528x³²+2173676076557926400x³⁴+4489970768427352064x³⁶-1782476573903945728x³⁸+2217097577445195776x⁴⁰+5478206734231142400x⁴²+8975251844884332544x⁴⁴+1480558377498050560x⁴⁶+4138526582576775168x⁴⁸+4512606826625236992x⁵⁰+3341670923508908032x⁵²-504403158265495552x⁵⁴+36028797018963968x⁵⁶
T₅₇(x)=57x-30856x³+4998672x⁵-383707584x⁷+17053670400x⁹-491145707520x¹¹+9848101109760x¹³-144438816276480x¹⁵+1605819780956160x¹⁷-13898323250380800x¹⁹+95567517969285120x²¹-530343064145756160x²³+2404221890794094592x²⁵-8986721141657698304x²⁷-9003663914688315392x²⁹+1573323872885800960x³¹+8888960991220465664x³³-7298749259682152448x³⁵+7268293646586216448x³⁷+5965341237748695040x³⁹+4962581960292564992x⁴¹-2735620114278776832x⁴³+3611253582453538816x⁴⁵+2443484272825204736x⁴⁷-3590213327944417280x⁴⁹+7415176786465521664x⁵¹+6931039826523193344x⁵³-1026820715040473088x⁵⁵+72057594037927936x⁵⁷
T₅₈(x)=-1+1682x²-470960x⁴+52559136x⁶-3123514368x⁸+114528860160x¹⁰-2831986360320x¹²+50104374067200x¹⁴-661377737687040x¹⁶+6717522904350720x¹⁸-53740183234805760x²⁰+344774682051870720x²²-1798824428096716800x²⁴+7715573085436379136x²⁶-8986404674508685312x²⁸+7566499303326220288x³⁰+842648647622459392x³²-2842498167826546688x³⁴-640725214082105344x³⁶-2127680206633172992x³⁸-8733159175657357312x⁴⁰+4446957186353987584x⁴²+4000252000267665408x⁴⁴+5741948787409027072x⁴⁶+748441963073634304x⁴⁸+6753710591195480064x⁵⁰-6958061424287416320x⁵²-4080261262397669376x⁵⁴-2089670227099910144x⁵⁶+144115188075855872x⁵⁸
T₅₉(x)=-59x+34220x³-5940592x⁵+488825856x⁷-23300699136x⁹+720203427840x¹¹-15512073830400x¹³+244647564410880x¹⁵-2928575256330240x¹⁷+27333369059082240x¹⁹-203047884438896640x²¹+1219892428249497600x²³-6001870746987528192x²⁵+5971123238820904960x²⁷-8969145434329055232x²⁹-4887069339942912000x³¹-7203663695975546880x³³+1613752924029059072x³⁵-8549744074750427136x³⁷+8226042422694510592x³⁹-3982156237897728000x⁴¹-6817209586722799616x⁴³+4389250418081792000x⁴⁵+9040413301992849408x⁴⁷+5087097254091685888x⁴⁹+6092244395925438464x⁵¹-2400418601388474368x⁵³-7133701809754865664x⁵⁵-4251398048237748224x⁵⁷+288230376151711744x⁵⁹
T₆₀(x)=1-1800x²+539400x⁴-64440320x⁶+4101166080x⁸-161130258432x¹⁰+4272393216000x¹²-81128521728000x¹⁴+1150672866508800x¹⁶-12574673417011200x¹⁸+108406921352970240x²⁰-750870450929664000x²²+4238609284595712000x²⁴-1272570505701883904x²⁶+2481907078440943616x²⁸-7058046098274779136x³⁰+7829956746201268224x³²+6881914849585004544x³⁴+3868231062140223488x³⁶+3474936130841870336x³⁸+6738499947336826880x⁴⁰+6035474411560108032x⁴²+812072899996286976x⁴⁴+3036552048754556928x⁴⁶-1114359432797487104x⁴⁸+3420483916987891712x⁵⁰+695806142428741632x⁵²-720575940379279360x⁵⁴+6269010681299730432x⁵⁶-8646911284551352320x⁵⁸+576460752303423488x⁶⁰
T₆₁(x)=61x-37820x³+7019392x⁵-617706496x⁷+31503031296x⁹-1042463944704x¹¹+24056860262400x¹³-406904607866880x¹⁵+5229920989347840x¹⁷-52482715893104640x¹⁹+419861727144837120x²¹-2721633330108825600x²³-3967654757530599424x²⁵-8516264250224672768x²⁷-4513784482498609152x²⁹+9217721217102905344x³¹+4416833114668531712x³³-6296667298568601600x³⁵-2160537874678677504x³⁷-1276170161010769920x³⁹-987587941138169856x⁴¹+441414336133464064x⁴³-2765104618089218048x⁴⁵-2967309204483735552x⁴⁷-7315816119686660096x⁴⁹+748723438050344960x⁵¹+3792030886245957632x⁵³+5692549928996306944x⁵⁵-1657324662872342528x⁵⁷+864691128455135232x⁵⁹+1152921504606846976x⁶¹
T₆₂(x)=-1+1922x²-615040x⁴+78479104x⁶-5336579072x⁸+224136321024x¹⁰-6357321105408x¹²+129242242252800x¹⁴-1964482082242560x¹⁶+23034515395706880x¹⁸-213372353139179520x²⁰+1590593905219338240x²²+8764868128896188416x²⁴-6662739009359314944x²⁶-1067691505180737536x²⁸-1969522866722439168x³⁰-7841258385705009152x³²+1951751379752058880x³⁴+1985178414432124928x³⁶-7796011880199225344x³⁸+9155903804351184896x⁴⁰-8010650293836447744x⁴²+70755772270641152x⁴⁴-8566761284932993024x⁴⁶-4820258976169984000x⁴⁸+394627917348339712x⁵⁰+801640733671948288x⁵²+8304637712871194624x⁵⁴+5116089176692883456x⁵⁶+5332261958806667264x⁵⁸+1152921504606846976x⁶⁰+2305843009213693952x⁶²
T₆₃(x)=-63x+41664x³-8249472x⁵+774664704x⁷-42176189440x⁹+1490736586752x¹¹-36771502473216x¹³+665389092372480x¹⁵-9158885153832960x¹⁷+98551746684518400x¹⁹-846606433423196160x²¹+5902821140547502080x²³+3050646941613424640x²⁵-4809213768493957120x²⁷+2378401472137134080x²⁹+5289977123161767936x³¹-1652605812368998400x³³-8246574015636832256x³⁵+6130894703542927360x³⁷+4130890474321870848x³⁹+852651476130988032x⁴¹+1984029149903192064x⁴³+2906616162630500352x⁴⁵+4280530708327301120x⁴⁷-2324701832653307904x⁴⁹+40532396646334464x⁵¹-2188749418902061056x⁵³-7530018576963469312x⁵⁵-6557241057451442176x⁵⁷-8646911284551352320x⁵⁹+1152921504606846976x⁶¹+4611686018427387904x⁶³