A Linear Recurrence With Polynomial Coefficients Satisfied by The Inegral of\ the, 2, -th Part of a Certain C-finite Polynomial Sequence By Shalosh B. Ekhad Proposition: Let , P[n](x), be the sequence of polynomial definied by the following recurrence P[n](x) = 2 x P[n - 1](x) - P[n - 2](x) Subject to the initial conditions P[0] = 1, P[1] = 2 x Equivalently, in terms of generating function infinity ----- \ n 1 ) P[n](x) t = -------------- / 2 ----- t - 2 t x + 1 n = 0 n 1 + (-1) n Define the Umbra, U, to be equal to, ---------, when applied to , x , n + 1 and extended linearly Also define Star(P)(x)=P(1/x)*x^n, where n is the degree of the polynomaial \ P(x) Let's define the sequence of numbers s(n), to be the action of U applied to, P[2 n](x) StarP[2 n](x) Then s(n) satisfies the following linear recurrence equation with polynomial\ coefficients 2 (4 n + 5) (4 n + 3) s(n) 2 (16 n + 48 n + 39) s(n + 1) ------------------------ - ------------------------------ + s(n + 2) = 0 (4 n + 9) (4 n + 7) (4 n + 9) (4 n + 7) Subject to the initial condition 26 526 s(1) = --, s(2) = --- 15 315 Just for fun, using this recurrence we get that s(1000) = 102159699098174036468403793521796424514700899135926217876197584188\ 130559831771070423787783838191009969389393523408379314967372304023147603\ 205519231140806561306975568151102293604028427004568198529983756197962600\ 277954975012387398626865325971124517525730174286860926696871376304992284\ 910065360823655740178973953386253779405514574796579889898461309777478181\ 786011968131789031259978354131679385208762883789235081040638647468180106\ 548924405515057182587952337205614469510994588797115281928612545662350860\ 221264275734411734714027511436569256656578726026029707861527382022655160\ 661911289972101269662055803298496778159779837666072243184242360925003545\ 057387064396476225469828169394831267859165176983603365230230529296037606\ 164523864677308219927618434820625023208160838175181674127766028973943581\ 945224967956980957461632839384865547491942860919275993529506678740943800\ 135400141011756371897507354624768224902585383751484360486622953826399153\ 079112021071905383222203253707880481027240148774242865473915061379125815\ 843005214065787793862539596887124667961014306043577526609066941671152862\ 607068890676880097019932112906401757791134356987471957030306614826593515\ 921698593656212243224154665236497074456779215709719561379673979965069467\ 612315247601470995161376915025590388580020840052137746500484036782062245\ 998820649964186811358241175798010797091012926428629323413230424851619462\ 143131738550379357857576109080454748535866016252785790939520943489937502\ 384148137548068149451841331201626115807611158084974183423839480645860142\ 244300115673634886995311834961846967198931362073613459135605388326949533\ 720721114235057388427314009469841375757449210329681655112905056055785872\ 793705845619404770451937802259300745712899604575181870501670377883013175\ / 9722481622 / 650265402624127211069120217493508230498306454837714837384\ / 817905856624734729012158011171846171935501306930541152788393476496644209\ 152676490220641223740289114493671707503261714533526884142890697191194486\ 730877356628198697500478240519569696884855057631052396025582963877499187\ 640728934247755219241749544518482404237423552841030138723722383094107262\ 639069781272591177642957389599641584071113236564973491371216714725273266\ 476173499162821306429991045335607168850178819530053866158787619700227585\ 769862783548554693304767527346617311231641754858636756968888380239503008\ 225493759446467102329019714173389107375222056971153907437893512665403160\ 829367000046935402050673864974005297774571037516951624087619061370364475\ 973746349632656981295701207857326917074525560584505835648306310770571293\ 387452879868888227320087760212587579575821889673911705134519402013137303\ 529769658142422799778431502897215881545068323525394112806417480263230608\ 761436784712815969419670463666622598473801755053376355339485611416247394\ 845203926567643182952328607141950767686313022334565608157029625798279713\ 170352273461978976513139299211294328103892845987019262716951852279373232\ 518657643952792231439904059403428697472386911287333648895895879952115755\ 847741271839805356354398498413387811785151625450438920476546858697836981\ 885077751104457104336359197890472581303885091615183607281944577270074905\ 795797059671403597224639876648145395203391521912944064914088651899757532\ 264764493412658207289960830362863494793143014855538928389175885436505262\ 904959058110891005146496683274737321887386282241662508713597456319652025\ 031305795830685962488658055213365279230308168210153469515638549590960434\ 711334134106478878193655417379807695801380353055563409254451087973007219\ 035903837334346875 This ends this article, that took, 0.138, seconds to generate. ---------------------------------------------------- Second part: i.e. Int(T_(2n)(x)*Star(T_(2n))(x),x=-1..1).. Try: ----------------------------------------------------------------------------\ ---------------- A Linear Recurrence With Polynomial Coefficients Satisfied by The Inegral of\ the, 2, -th Part of a Certain C-finite Polynomial Sequence By Shalosh B. Ekhad Proposition: Let , P[n](x), be the sequence of polynomial definied by the following recurrence P[n](x) = 2 x P[n - 1](x) - P[n - 2](x) Subject to the initial conditions P[0] = 1, P[1] = x Equivalently, in terms of generating function infinity ----- \ n -t x + 1 ) P[n](x) t = -------------- / 2 ----- t - 2 t x + 1 n = 0 n 1 + (-1) n Define the Umbra, U, to be equal to, ---------, when applied to , x , n + 1 and extended linearly Also define Star(P)(x)=P(1/x)*x^n, where n is the degree of the polynomaial \ P(x) Let's define the sequence of numbers s(n), to be the action of U applied to, P[2 n](x) StarP[2 n](x) Then s(n) satisfies the following linear recurrence equation with polynomial\ coefficients 2 (4 n + 3) (2 n + 1) (4 n + 1) (20 n + 61 n + 46) s(n) -1/4 ------------------------------------------------------ 2 (4 n + 7) (2 n + 3) (4 n + 9) (20 n + 21 n + 5) 5 4 3 2 (1920 n + 4256 n - 3496 n - 13834 n - 9621 n - 1890) s(n + 1) - 1/4 ----------------------------------------------------------------- 2 (4 n + 7) (2 n + 3) (4 n + 9) (20 n + 21 n + 5) + s(n + 2) = 0 Subject to the initial condition -22 254 s(1) = ---, s(2) = --- 15 315 Just for fun, using this recurrence we get that s(1000) = 145406848008123276029375524606533673582715376817834822962293182555\ 727398747332311717785628481355932272950238554220842768566842642173167628\ 140164642459890581564892431329700100187182580793744663225459742534745802\ 782889511743874605055550228230726069500009144949163952552993267820501111\ 072258674096722471849368596703734083725255071622730703671839810789043588\ 786514638905616066749866967308409997475695814987182035152116995796862295\ 930873434094907428357477747514075120101549915269231112456674591388401550\ 302245622585905125486447556998108385058896958985660146628828331484141529\ 375154241308296378035070853175998758302044286750570109717213175050370454\ 108134899477474803721411942757686700640179527438952664124482682198521974\ 238925844912104627740004198694104104060952186839608953862244598498026477\ 769206987656245372969199561816781454993968807191114513478134061644879048\ 388004861189594649502372849079801745459738285698530885077138911854069541\ 471106239796882907880125741148005636848972107093566090068512114261663956\ 751766212613583539912188948683020842480455833048061732478325396047131683\ 053493623225583061022002761633076184667871468709846309534690758189046323\ 227245387791721172786524888531471213041115915331914241072425706697327364\ 872750667409206081366098479971921904549511891644766438288364199958021625\ 923107100086516094521315711244834615758583973557420524055222633862463092\ 251499068711685094281273821004533447421869073391581657203159929246421181\ 841247860561232129295373803215376707357247335550347520990365619581998120\ 827028301549489296647287996566147640903686117560284189900389691841540046\ 420764237228953751613493543001057999107581999084304474894019268011717075\ 379836021557833641256680694151177868049763038392347514816705373684642918 / / 7751163599279596355943912992522618107539812941665560861627029437810\ / 966837969824923493168406369471175578612050541237650239839998973099903763\ 430043386984246244764566753438879637239640458983257110519038281832058091\ 008128474205700626993270786867472286962144560624948929419790316677488896\ 233242213361654570660310258510088749865079253586770806481758570657711792\ 769286837504052084027727682127669779854484017144903239525257336395988110\ 020829972645493260400437452694131528798242084612748426826712822376764379\ 898771944192828925971678349881169717914950143069149492454875858047885612\ 601887859761914992946798159912646919096154576659690670971605677086054640\ 559469992444032470490143149472886767202063359124419211534744553607056487\ 621271217044758397659336851528344682167309560927811224385209817178438328\ 037147669655446101734043948543796924913027525203471271996596658074854325\ 057679773358903514534813308017214416422697824652496364737708856436326473\ 776766355482471926906141373807716920236246155646668488081668946554830804\ 686306740791756997132053150820851226228022049231793139515494180990599099\ 666789400036620446598628390998402724165269611586066079170128931622399115\ 917283398763656388088870073870851982545017094839078889029219809705075960\ 330479847744430101087582716479007375369231932080438555678216824070126793\ 165128683689401638854433169142310292052988598800779361059292877085900951\ 283130878917707329645893110824426941202293253775936730645109784595992761\ 478885830896333097925332857934264737078024026398976554403142733827111972\ 681820781346240464634868876897644484320617103866081679330252138373165086\ 301776672864804018143314128425273365065029356626411511124248381759102878\ 549228228068372575167307733952453808422315838313056968638246050907973741\ 02541475 This ends this article, that took, 0.315, seconds to generate. ---------------------------------------------------- For Theorem 2 of that paper, First part: ----------------------------------------------------------------------------\ ---------------- A Linear Recurrence With Polynomial Coefficients Satisfied by The Inegral of\ the, 2, -th Part of a Certain C-finite Polynomial Sequence By Shalosh B. Ekhad Proposition: Let , P[n](x), be the sequence of polynomial definied by the following recurrence P[n](x) = 2 x P[n - 1](x) - P[n - 2](x) Subject to the initial conditions P[0] = 1, P[1] = x Equivalently, in terms of generating function infinity ----- \ n -t x + 1 ) P[n](x) t = -------------- / 2 ----- t - 2 t x + 1 n = 0 n (1 + (-1) ) binomial(n, n/2) Define the Umbra, U, to be equal to, 1/2 ----------------------------, n 2 n when applied to , x , and extended linearly Also define Star(P)(x)=P(1/x)*x^n, where n is the degree of the polynomaial \ P(x) Let's define the sequence of numbers s(n), to be the action of U applied to, P[2 n](x) StarP[2 n](x) Then s(n) satisfies the following linear recurrence equation with polynomial\ coefficients 1/4 s(n) + s(n + 1) = 0 Subject to the initial condition s(1) = -1/4 Just for fun, using this recurrence we get that s(1000) = 1/1148130695274254524232833201177681984022317702088695200477642736\ 825766261392370313856659486316506269918445964638987462773447118960863055\ 331425931356166653185391299891453122800006887791482400448714289269900634\ 862447816154636463883639473170260404663539709049965581623988089446296056\ 233116495361642219703326813441689089844585056023794848079140589009347765\ 004290027167066258305220081322362812917612678833172065989953964181270217\ 798584040421598531832515408894339020919205549577835896720391600819572166\ 305827553804255837260155283487864194320545089152757838826251754355288008\ 22842770817965453762184851149029376 This ends this article, that took, 0.059, seconds to generate. ---------------------------------------------------- Second part: ----------------------------------------------------------------------------\ ---------------- A Linear Recurrence With Polynomial Coefficients Satisfied by The Inegral of\ the, 2, -th Part of a Certain C-finite Polynomial Sequence By Shalosh B. Ekhad Proposition: Let , P[n](x), be the sequence of polynomial definied by the following recurrence P[n](x) = 2 x P[n - 1](x) - P[n - 2](x) Subject to the initial conditions P[0] = 1, P[1] = 2 x Equivalently, in terms of generating function infinity ----- \ n 1 ) P[n](x) t = -------------- / 2 ----- t - 2 t x + 1 n = 0 n (1 + (-1) ) binomial(n, n/2) Define the Umbra, U, to be equal to, 1/2 ----------------------------, n 2 (n + 2) n when applied to , x , and extended linearly Also define Star(P)(x)=P(1/x)*x^n, where n is the degree of the polynomaial \ P(x) Let's define the sequence of numbers s(n), to be the action of U applied to, P[2 n](x) StarP[2 n](x) Then s(n) satisfies the following linear recurrence equation with polynomial\ coefficients 1/4 s(n) + s(n + 1) = 0 Subject to the initial condition s(1) = -1/8 Just for fun, using this recurrence we get that s(1000) = 1/2296261390548509048465666402355363968044635404177390400955285473\ 651532522784740627713318972633012539836891929277974925546894237921726110\ 662851862712333306370782599782906245600013775582964800897428578539801269\ 724895632309272927767278946340520809327079418099931163247976178892592112\ 466232990723284439406653626883378179689170112047589696158281178018695530\ 008580054334132516610440162644725625835225357666344131979907928362540435\ 597168080843197063665030817788678041838411099155671793440783201639144332\ 611655107608511674520310566975728388641090178305515677652503508710576016\ 45685541635930907524369702298058752 This ends this article, that took, 0.074, seconds to generate. ----------------------------------------------------