The following are five random examples of pure scheme of the Taylor coefficients of functions with one variable --------------------------- A Pure Recurrence Scheme that enables Linear-Time and Constant-Space Calcula\ tion of the Taylor coefficients of the function with, 1, variables , x[1] 4 3 2 (1/3) (-95 x[1] - 56 x[1] + 21 x[1] - 8 x[1] + 1) By Shalosh B. Ekhad m[1] Theorem: Let , F(m[1]), be the coefficient of, x[1] , in the Taylor expa\ nsion (around the origing) of the multivariable function (that may also \ be viewed as a formal power series) of, x[1] 4 3 2 (1/3) (-95 x[1] - 56 x[1] + 21 x[1] - 8 x[1] + 1) The following PURE recurrence relations hold in each of the, 1, discrete variables, m[1] (3 m[1] - 4) F(m[1] - 1) 7 (3 m[1] - 8) F(m[1] - 2) F(m[1]) = 8/3 ------------------------ - -------------------------- m[1] m[1] 56 (m[1] - 4) F(m[1] - 3) (3 m[1] - 16) F(m[1] - 4) + ------------------------- + 95/3 ------------------------- m[1] m[1] and in Maple notation F(m[1]) = 8/3*(3*m[1]-4)/m[1]*F(m[1]-1)-7*(3*m[1]-8)/m[1]*F(m[1]-2)+56*(m[1]-4) /m[1]*F(m[1]-3)+95/3*(3*m[1]-16)/m[1]*F(m[1]-4) Subject to the initial conditions -1048 F(0) = 1, F(1) = -8/3, F(2) = -1/9, F(3) = ----- 81 -------------------------------------------- By the way, it took, 0.111, seconds to geneate the scheme. -------------------------------------------- Using this scheme we can compute any value very fast. For example F(1000) = -21521766491443053346885317873121155574755522697241061954043153627\ 807041779220596699068654875657689673246378621491030277623641299668727157\ 801083644956528474616642655538783130441220529467463279913204023426692066\ 793660950554806480614485159438013028993561071903909684196950937545386344\ 089031968114008103935665215593136264148265158568209392372997721372960727\ 565983679343407312573284971664158200927153321412837129148634989598729761\ 525238832625606952237893956524988669932370894112865914908447818621375788\ 057099304053444949458396946288696173458255547981358322924390613154163025\ 212853515232358853637304993625949521426089874201518401822820761852770002\ 655136110142798837866861552848886477967141170181767414735273726496996631\ 529780082135560024952744766246345108589572447931224591934433966441028200\ 315623755558768389899882177643368513848454013934202120322972978405765998\ 891124621811672775569662691445589724538006370987678042687286136308317874\ 373398881463928718000253008489936913576710650146579031149065521079624989\ 145382536616439359614723807830403607755733084013908881534256253689735999\ 713106257148372601104911463518308835926202541573103838773494802597116772\ 309321172944857261770386105773634583528785374261552207575989382675227589\ 571637135369662343483442779312263055698975528780916810146313167130412317\ 177540852738954272061927397318493858719898352200593339450993946645321911\ 767341838540606644721480894803479539738514146021140787334077630400355694\ 829786922149617462756865563770314166035736950515759153625089662861086297\ / 504398884140536 / 5341208974569668648157600007095602765915827471658279\ / 432881611524166793699657977262077686251757608568524753819549730954511886\ 572669352429351814397201942588209003190059552575219955135525776835647834\ 966041641308292978654521602542482600658200053877960401970066291044955655\ 225128887435055704499238108359363624230061338433101247330137630242204383\ 644772105699157258332633943511440449655520801797701667919843238647929827\ 475182398546217736244883384277009182128250675825710009967536673771014122\ 159905552997954403447587192744788679886342111360648430081695351811681596\ 198457201492321756577412484860206561422528082272505113129035094264821529\ 968932061315087366071771309053414973090377237712900231649897711624396664\ 440924714758889 This took, 0.087, seconds. --------------------------- A Pure Recurrence Scheme that enables Linear-Time and Constant-Space Calcula\ tion of the Taylor coefficients of the function with, 1, variables , x[1] 4 3 2 (1/3) (-60 x[1] - 38 x[1] - 17 x[1] + 43 x[1] + 1) By Shalosh B. Ekhad m[1] Theorem: Let , F(m[1]), be the coefficient of, x[1] , in the Taylor expa\ nsion (around the origing) of the multivariable function (that may also \ be viewed as a formal power series) of, x[1] 4 3 2 (1/3) (-60 x[1] - 38 x[1] - 17 x[1] + 43 x[1] + 1) The following PURE recurrence relations hold in each of the, 1, discrete variables, m[1] (3 m[1] - 4) F(m[1] - 1) (3 m[1] - 8) F(m[1] - 2) F(m[1]) = -43/3 ------------------------ + 17/3 ------------------------ m[1] m[1] 38 (m[1] - 4) F(m[1] - 3) 20 (3 m[1] - 16) F(m[1] - 4) + ------------------------- + ---------------------------- m[1] m[1] and in Maple notation F(m[1]) = -43/3*(3*m[1]-4)/m[1]*F(m[1]-1)+17/3*(3*m[1]-8)/m[1]*F(m[1]-2)+38*(m[ 1]-4)/m[1]*F(m[1]-3)+20*(3*m[1]-16)/m[1]*F(m[1]-4) Subject to the initial conditions 409667 F(0) = 1, F(1) = 43/3, F(2) = -1900/9, F(3) = ------ 81 -------------------------------------------- By the way, it took, 0.051, seconds to geneate the scheme. -------------------------------------------- Using this scheme we can compute any value very fast. For example F(1000) = -21601242164887284473098743717193326932546325642401633458410105547\ 597442999469758393090034067503490108807183807233383595138934140412496046\ 603729849328740155792182198548400195578880189162594973548502789715439296\ 195989671380835749749602238479275319622793071265124483935264160035677545\ 734998051649065901229733613818591247674531303090305970196478949506517627\ 020193930319910168106368249455470050041665966132265467625408896843091940\ 275729801112583434296258116086284683364907432303654061786559938938456758\ 209176154794230571683643322663445673040292240866206253111222181429256383\ 816428421271654549754092127108341522973196930585165257637365005774670903\ 148083654161510281520793650713364865544288959340762034134649621401613162\ 832162731244838985191065447832382562265662925809236589104411425930325802\ 345046312837518167859460199952542509455477320162006314166948699725851602\ 061456519621794363145661603001064514596606583547705506680175218588077384\ 940923194770241785742421579395592477444549887150107731819238030117574583\ 713360965680347389912065490539337133864440647963185558611947662323518562\ 032334277960424053513621416652395063766116674664377197511885184149596130\ 114970845835434418278537298124541689456788405161483941816382998166432368\ 199221878328170356699558059448299862229668455067478753289319508442269696\ 351015979195566445900553966380911555457757652096905220731138414251238581\ 988671904298935462864084018340947260550543659867203223393024915742251442\ 642332418951152358797185648605372553444706332728810068532147307412644711\ 143357733695513582927236654866177733850407715027039981365410643180158223\ 547745749480815113174043808892987374684714768198525616029879717384219556\ 109423955161028205686464626959575862914078362362885879402105925252114186\ 047160209874919699208653156440733292933478633235036214184916950490506378\ 165053535146925647582518322405765118609405806549110644700518709194011126\ 152328597375304218485553853242872968977844869499781526071684245147322487\ 851011037603882163198684735765037045644633861992496560313198424508858992\ 957481762581165037276061639950916920023589608712664204061581975704963439\ 613543935459107419994613165011603866561659290389706396672870647747055716\ 993210438791062374077352244545080092863217068583541697385834753567424370\ 273338143055829585327198887699407811381633964776754820266404446415113171\ / 232133348630227793823777980847948290043367336544108 / 5341208974569668\ / 648157600007095602765915827471658279432881611524166793699657977262077686\ 251757608568524753819549730954511886572669352429351814397201942588209003\ 190059552575219955135525776835647834966041641308292978654521602542482600\ 658200053877960401970066291044955655225128887435055704499238108359363624\ 230061338433101247330137630242204383644772105699157258332633943511440449\ 655520801797701667919843238647929827475182398546217736244883384277009182\ 128250675825710009967536673771014122159905552997954403447587192744788679\ 886342111360648430081695351811681596198457201492321756577412484860206561\ 422528082272505113129035094264821529968932061315087366071771309053414973\ 090377237712900231649897711624396664440924714758889 This took, 0.075, seconds. --------------------------- A Pure Recurrence Scheme that enables Linear-Time and Constant-Space Calcula\ tion of the Taylor coefficients of the function with, 1, variables , x[1] 4 3 2 (1/3) (5 x[1] + 2 x[1] + 38 x[1] - 52 x[1] + 1) By Shalosh B. Ekhad m[1] Theorem: Let , F(m[1]), be the coefficient of, x[1] , in the Taylor expa\ nsion (around the origing) of the multivariable function (that may also \ be viewed as a formal power series) of, x[1] 4 3 2 (1/3) (5 x[1] + 2 x[1] + 38 x[1] - 52 x[1] + 1) The following PURE recurrence relations hold in each of the, 1, discrete variables, m[1] (3 m[1] - 4) F(m[1] - 1) (3 m[1] - 8) F(m[1] - 2) F(m[1]) = 52/3 ------------------------ - 38/3 ------------------------ m[1] m[1] 2 (m[1] - 4) F(m[1] - 3) (3 m[1] - 16) F(m[1] - 4) - ------------------------ - 5/3 ------------------------- m[1] m[1] and in Maple notation F(m[1]) = 52/3*(3*m[1]-4)/m[1]*F(m[1]-1)-38/3*(3*m[1]-8)/m[1]*F(m[1]-2)-2*(m[1] -4)/m[1]*F(m[1]-3)-5/3*(3*m[1]-16)/m[1]*F(m[1]-4) Subject to the initial conditions -667418 F(0) = 1, F(1) = -52/3, F(2) = -2590/9, F(3) = ------- 81 -------------------------------------------- By the way, it took, 0.076, seconds to geneate the scheme. -------------------------------------------- Using this scheme we can compute any value very fast. For example F(1000) = -75377727986285887279741212064668352128204182069789487903958360766\ 870687974614791308832859745567030558672794987851211380175677481074177952\ 753953313816245840510336580372870260335498133903527541363697503884185445\ 882979476555190735119641480100717932933776396420201036461361106821188876\ 811581123294692638221284051226105521659493359646809338488520237131768553\ 861080836564870553251490563737324895532011886162321044700526956129169922\ 187813434554213524089321417904192430136061190123215451261533609190008424\ 358003433912519506668117408177862229880798092367113593692515661780798889\ 578618952741401702899795623260594658243853631328255324507247958703169397\ 256699077724639365516991629163502604417513144502381253888955199154819809\ 966710615160058105269575923914267217301430544385624685510860521465500596\ 516638117398053542760501362545535411453408518559052637737280603061517779\ 868910427116692673422969038166707346078859959130466368759536564924936355\ 208787189149431135389936197093137399193909014524312062791404081696820776\ 321300695206401430737733689690728494349126118080373168390974076976663168\ 629190428142077228159309568519959597170180191013789764293394883470112574\ 899013641305965419103368547525269573236090905000678096180921931580735258\ 526114293207051359936312971250138759254297880862657722639523873289668448\ 001455653577781172684389032106491869666972157656818830954986936888119257\ 028220126842785617165458330812193041427960253841342841160692449691955897\ 183566880305981675530096439472419232825639079424806832688359297853460780\ 675967526212487361710665901841403350652667509998041944287150856868598705\ 332194792821674370826536805099775176028467249975916899173926260453673774\ 397289841560124254329598586487883200958494389846867843261914562016498615\ 836142705980011798708315526572023710204616768517218533692224288402295837\ 540137110578292096366072472841874989004011066291916713956604508136970141\ 312611853423967389907387994404794160516138515648516464342303892306809116\ 443388861485353032961561118429795126197742352775643793034403586519000550\ 878572593714278194422838235432925748538938053254073312467355275762486625\ 677540242589385017302383365205975855809599551547313654428644541178077825\ 962156021968708995499907966011313295871933718757948407239170695768239889\ 900454770290673932032925865949153446959106945882862884453270348561453684\ 247197706736401432510223940729688567915969851558819203390385611631805055\ / 045345668910378979507584313427459893632741257299808 / 5341208974569668\ / 648157600007095602765915827471658279432881611524166793699657977262077686\ 251757608568524753819549730954511886572669352429351814397201942588209003\ 190059552575219955135525776835647834966041641308292978654521602542482600\ 658200053877960401970066291044955655225128887435055704499238108359363624\ 230061338433101247330137630242204383644772105699157258332633943511440449\ 655520801797701667919843238647929827475182398546217736244883384277009182\ 128250675825710009967536673771014122159905552997954403447587192744788679\ 886342111360648430081695351811681596198457201492321756577412484860206561\ 422528082272505113129035094264821529968932061315087366071771309053414973\ 090377237712900231649897711624396664440924714758889 This took, 0.073, seconds. --------------------------- A Pure Recurrence Scheme that enables Linear-Time and Constant-Space Calcula\ tion of the Taylor coefficients of the function with, 1, variables , x[1] 4 3 2 (1/3) (-77 x[1] - 37 x[1] - 56 x[1] + 26 x[1] + 1) By Shalosh B. Ekhad m[1] Theorem: Let , F(m[1]), be the coefficient of, x[1] , in the Taylor expa\ nsion (around the origing) of the multivariable function (that may also \ be viewed as a formal power series) of, x[1] 4 3 2 (1/3) (-77 x[1] - 37 x[1] - 56 x[1] + 26 x[1] + 1) The following PURE recurrence relations hold in each of the, 1, discrete variables, m[1] (3 m[1] - 4) F(m[1] - 1) (3 m[1] - 8) F(m[1] - 2) F(m[1]) = -26/3 ------------------------ + 56/3 ------------------------ m[1] m[1] 37 (m[1] - 4) F(m[1] - 3) (3 m[1] - 16) F(m[1] - 4) + ------------------------- + 77/3 ------------------------- m[1] m[1] and in Maple notation F(m[1]) = -26/3*(3*m[1]-4)/m[1]*F(m[1]-1)+56/3*(3*m[1]-8)/m[1]*F(m[1]-2)+37*(m[ 1]-4)/m[1]*F(m[1]-3)+77/3*(3*m[1]-16)/m[1]*F(m[1]-4) Subject to the initial conditions 113089 F(0) = 1, F(1) = 26/3, F(2) = -844/9, F(3) = ------ 81 -------------------------------------------- By the way, it took, 0.049, seconds to geneate the scheme. -------------------------------------------- Using this scheme we can compute any value very fast. For example F(1000) = -44875577020788048635242090008772579163199848221450965791616699845\ 997028485088936380449597226237877604211449392237451002980254248303310361\ 624221660692862294546947362531977745087975262153496374956926135124293888\ 848263995980614865650032782768932479660208231576390322994866121229574692\ 234922706931770697220137030454753013861018140208672102681417926985162217\ 134531639349861474606874979918433757336472477340700900503542139619162815\ 089172802458420676169733606236375538711597870634887584424764966317323455\ 612725470699098584097744012826602128087942349617522247281518441497194155\ 683017803797180775253870986077978019154440446488019671379896555504147760\ 489640273775623487868081183534593725717882382775413445762153185964722395\ 603501970297215752655482043533172113773610378061723979693614138245462994\ 781905443207068593696688977887117692257963927587505627403505006453597410\ 495162581326964430828371256284247797404110424599246305277924722004915913\ 747099644790333260177786917795117564044416974532498278036543487235609618\ 562303829003311092795605032952876291646830439492612405010824991495554803\ 549909883338245521148438762417728693298711247269837003010997252113043965\ 606030170264859034986514165758405145549303774884132341055726499224196902\ 241290642899133518137317479211017632902481007664677057428362640857147346\ 029541344438909354251279550913760003033386067509435598300062075282186752\ 592092789840290436917219120178527164811345240473644739871657256224342652\ 753947614067881499440710836995027479463208434417089057485310386639841338\ 813437520692268196506615848530146363412215048456374940076157977986522596\ 247444658973321497973367433283484603391784959476757820134628954310521411\ 195208276803963789234046812638140350867855503551069254021795074107691493\ 435298993054258261010865162514460327924556883939582899638773733866319340\ 739457215565778744041444840374975211265341845497804629559260541252699150\ 131033754590930441664990043190221316887907216105118573393954406007573884\ 591748623119638740999538646029162007812768351528354368980170751497256283\ 850536629769323664289012790342279705012272015059371350302158822884711947\ 484746043117285269529024423094823361604227238747456157781906891848107841\ / 4208 / 534120897456966864815760000709560276591582747165827943288161152\ / 416679369965797726207768625175760856852475381954973095451188657266935242\ 935181439720194258820900319005955257521995513552577683564783496604164130\ 829297865452160254248260065820005387796040197006629104495565522512888743\ 505570449923810835936362423006133843310124733013763024220438364477210569\ 915725833263394351144044965552080179770166791984323864792982747518239854\ 621773624488338427700918212825067582571000996753667377101412215990555299\ 795440344758719274478867988634211136064843008169535181168159619845720149\ 232175657741248486020656142252808227250511312903509426482152996893206131\ 508736607177130905341497309037723771290023164989771162439666444092471475\ 8889 This took, 0.097, seconds. --------------------------- A Pure Recurrence Scheme that enables Linear-Time and Constant-Space Calcula\ tion of the Taylor coefficients of the function with, 1, variables , x[1] 4 3 2 (1/3) (46 x[1] + 21 x[1] - 2 x[1] + 93 x[1] + 1) By Shalosh B. Ekhad m[1] Theorem: Let , F(m[1]), be the coefficient of, x[1] , in the Taylor expa\ nsion (around the origing) of the multivariable function (that may also \ be viewed as a formal power series) of, x[1] 4 3 2 (1/3) (46 x[1] + 21 x[1] - 2 x[1] + 93 x[1] + 1) The following PURE recurrence relations hold in each of the, 1, discrete variables, m[1] 31 (3 m[1] - 4) F(m[1] - 1) (3 m[1] - 8) F(m[1] - 2) F(m[1]) = - --------------------------- + 2/3 ------------------------ m[1] m[1] 21 (m[1] - 4) F(m[1] - 3) (3 m[1] - 16) F(m[1] - 4) - ------------------------- - 46/3 ------------------------- m[1] m[1] and in Maple notation F(m[1]) = -31*(3*m[1]-4)/m[1]*F(m[1]-1)+2/3*(3*m[1]-8)/m[1]*F(m[1]-2)-21*(m[1]-\ 4)/m[1]*F(m[1]-3)-46/3*(3*m[1]-16)/m[1]*F(m[1]-4) Subject to the initial conditions F(0) = 1, F(1) = 31, F(2) = -2885/3, F(3) = 49700 -------------------------------------------- By the way, it took, 0.048, seconds to geneate the scheme. -------------------------------------------- Using this scheme we can compute any value very fast. For example F(1000) = -24850645288429253578480526705231137216703223814521007530218937555\ 424865144327407856190805635061657466811165247314923422950737748602267978\ 124399867009498755848945778357197216625272853755067209280014896784554058\ 285650614258145526205774604968558819963891213103443515479472556112297402\ 013161616964365805987992814895049927683789336830549938716910962552529954\ 129965646874326489058501130808231563760334766314686382246015552081594583\ 857312365689426453716037778679947995675918677428044565545496813142452458\ 982785833610170501044600553822708093672096295234998769450929283877126958\ 529310420734611343003757975766690909201785370646306909799200440716222525\ 944998803897875565802842085989936096272591660760734229458265863827399776\ 740883016234698584325545080198600129574467461184009682412741487249353706\ 230128385350014948872652144525905232437215363016128577914783991818398089\ 779944906679282894997162031089924992282408366827483534657622105164680909\ 902727491311215208849557608231121988729732201871144112527332944138944632\ 415422406853204791926140928358513396487211438817083950051857745427623026\ 942985178584650368521182822073259818958140530101731555553322770618367829\ 006260946681468231503683451601829180351560241264672491561786089121725176\ 136368079046717964997960481099563501063782237023141312890260503498945479\ 481324268804762994484693233197243753418235520303328921597759680446522851\ 815968477398748822563411783924312947633137603153171802733428724464380287\ 670228467728580881843999978470331967908172014207641855324100448733321712\ 366723667262482948604786039294687998722044442654634699761513356737073916\ 621579200175843600026436003972206848059780842830578130721312996842422527\ 693246334258102182001945026823252498403967635559214914619768573381467093\ 994689635129949461800429823443768731376514580578026914599291907914405744\ 435342296738415447132658588873456659888461478978921698595374431280580814\ 388816226713114223366937998328355045740646528246070561023950996435391879\ 459078583814413124852751383586383075160872976695745299837721568684939870\ 057719482179520523058916131459620222675720193333200447037056471149619690\ 727430616607720891502822126637111908323750332374078727677665443918981869\ 082081037318300186870567446864892179926763934077036470362385309342987803\ 951218360320681091095569432353815397340672445978404359728290693195533666\ / 276584746470981511981997 / 2567895082317366481941951864835043450789422\ / 772074019578389344849699428653496003365116248323641839386088990301221856\ 382418166468700571820999743261827854906636621247382689843970890332346792\ 903307890170455921369484306506016269454180344229084589175822697179805191\ 122949699685981065094664629515287699202229468556495525602304313520992006\ 95403082087370528607985787 This took, 0.056, seconds.