A TREATY OF SYMMETRIC FUNCTION PART 1
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- Charles Casey
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1 A TREATY OF SYETRIC FUNCTION AN APPROACH IN DERIVING GENERA FORUATION FOR SUS OF POWER FOR AN ARITRARY ARITHETIC PROGRESSION PART ohd Shu Abd Shuo Soc Ieol Sd hd - Jl He Jl Reo Kg Selgo ly gezv@yhoo.co.u Su of Powe hd gheed ee of y clcl hec fo oe h wo houd ye go. The ue of fdg u of owe o dcee u of uecl owe c be ced bc fo he e of Achede hd C he o Fulhbe he ee ceuy. Ul ody hee o cloed fo u of owe foulo fo hec ogeo h bee foud. y hec wee volved h eech d y oche hve bee oduced bu oe foud o be cocluve. The geelzed euo fo u of owe dcoveed h eech h bee coed o Fulhbe u of owe fo ege d foud h h ew geelzed euo c be ued fo boh ege d hec ogeo hu offeg ew foe udyg yec fuco Fe l heoe Re Ze fuco ec. Keywod: Su of Powe Fulhbe u of owe yec fuco Su of Nuecl Powe Powe Su. Scoe: Pue hec
2 Ioduco. The u of owe h gheed he ee of y hec ce he ce e ul ody. The u of ege fo e w fouled by Pyhgo [] -C d he foulo gve follow: [] Whle he u of ue fo ege foulo w dcoveed by Achede [] -C h foulo gve follow: [] The u of cube w f fouled by Id hec by he e of Aybh who w bo [] h u of cube foulo gve follow: [] Ohe hec uded h foulo wee Abu Al-K [] - d ev be Geo [] -. The u of fouh owe of ege w fouled by Abu Al Al-H b Al-H b Al- Hyh [] - whle he w Egy h foulo c be ee follow: [] Su of owe w f oduced o wee wold by Tho Ho [] -. H wo coceed o he u of ue cube d fouh owe. The u of owe fo ege fo hghe owe wee fouled by Fulhbe []. He woed u euo fo u of owe fo ege u o h owe. Howeve D.E. Kuh [] h eoed h Acde Algeb Fulhbe ged o foule euo fo u of owe u o d owe. The geel foulo of u of owe fo hec ogeo ly fo ege w fouled by le Pcl [] d w we wod h boo Te du Tgle. The eloh of h euo gve follow: [] Th foul c be fuhe we follow: [] I Job eoull h boo A Coecd whch w ublhed ohuouly deved he ybolc geel foulo fo u of owe fo ege whch e he couo ug geelzble foulo oble []. The geelze foulo gve follow: y dog Fulhbe heoe. Wll e l [] dcoveed he foul fo u of odd owe by codeg hec ogeo of he fo follow: K []
3 e he u of hec ee euo [] follow: λ [] The u of owe fo h ogeo gve follow: S K [] Fo oe odd.e. - he foul fo u of owe e gve follow: S [ λ] [ λ] S [ λ] [ λ] [ λ] [] [] S [ λ] [ λ] [ λ] [] Adog Yoh Ib ehod [] fo coug he -h u of owe fo he f e of hec ogeo N. Guhe [] deved foul fo coug he u of -h owe of ucceve e of hec euece. The foulo gve follow: e he u of owe o hec e follow: S b b b b [] H eul fo gve follow: S [] The ech of le geel foulo fo u of owe fo hec ogeo hd ced y hec d dffee ehod hd bee ooed o eee he uo fo ye []- []. Th e o ee eleg ehod fo he u of owe of -h fo f e of hec ogeo. The uoe of h ehod o couc le euo. A Aleve Devo d Foulo of he Su of Powe fo -h Ahec Pogeo. The de of h e o ed he u of owe e o bc yec fuco wh eeou coeffce. The geelzed euo ooed o be he fo follow: u [] Whee: f eve f odd coeffce d fo _ odd _ d u [] fo _ eve _ y edg he geel euo [] fo f yeld
4 [] [] [] [] [] [] [] [] [] []
5 . D Aly ehod. Th ehod bou d ly d ug he eul o couc he euo eeded fo ech of -h e. e h euo below le: b [] Tbulg oe vlue of fo d yeld: Tble D fo Su Su _ Tble D fo Su Su _ y codeg evel vlue of d log Su veu Su fo hee vlue of yeld he gh follow:
6 SuX^ V SuX Wh Vou Vlue Of y.sux^ y.sux^ SuX^ y.sux^. y.sux^ SuX^_ SuX^_ SuX^_ SuX^_ SuX Fgue. The cuve fo Su^ veu u.
7 y collecg he coeffce of d b fo ech d bulg he d log he yeld Fgue. d Fgue.. Tble Coeffce fo d b vou b Fgue. Cuve fo veu. Fgue. Cuve fo b veu. Theefoe lfyg ll he coeffce fo u of owe fo yeld: [] Th euo oly lcble fo he ege. The euo fo u of owe fo by hec ogeo fo c be obed by bulg he d of he hec ogeo wh dffeece.
8 Now code h euo: u [] y vyg he vlue of d g he vlue fed.e. d bulg he d fo vou vlue of yeld Tble o Tble. Tble Tbuled d fo. Su Su Tble Tbuled d wh Su Su Tble Tbuled d wh Su Su
9 y log he d fo he Tble o Tble yeld he cuve Fgue fo vou vlue of. SuX^ V SuX Wh Vou Vlue Of Ad SuX^ y.sux^ SuX^ SuX^ SuX^ SuX^ y.sux^ y.sux^. y.sux^. SuX Fgue. The cuve fo Su veu Su fo vou vlue of d. Tble Coeffce d u vou vlue of wh. u u u c c c
10 Sce c u d ubug h vlue o euo [] yeld: [] Th d ly ehod c be eeed fo gve follow: Fo eve K [] K [] K [] K [] I he fo: [] Solvg fo he coeffce yeld: [] y vyg he vlue of we ge vou vlue of coeffce K he fo of. Fo odd K [] K [] K []
11 K [] I he fo: [] Solvg fo he coeffce yeld: [] Ag by vyg he vlue of we ge vou vlue of coeffce K he fo of. Th ehod c be ued o geee hec -h e fo y vlue of. Howeve he lge he vlue of he oe edou he clculo would be. Sce coof Ecel hvg u eco of dg he eo clculo wll occu fo ube oe h dg. I ode o ovecoe h oble ddo hould be lled o he coof Ecel h eech w doe ug Xube [] whch led o eceo of u o dg.. Algebc ulo ehod. Fo ll he u of owe c be deved ug le lgebc ulo of hec e. The foulo fo oe ll c be obed follow: Fo d e [] d [] Sce he ee hec ogeo hu [] Subug [] o [] yeld [] Subug [] o [] yeld [] Regg [] yeld []
12 o [] Now code d [] Sce eeg fo e yeld [] Theefoe [] Sce he d [] ddg [] d [] yeld [] ug boh de [] d egg yeld [] Subug [] [] d [] o [] yeld [] egg [] yeld o [] [] eeg he e ocedue fo e fo o d by codeg geel foulo fo of h fo b d he clculg fo oe d bulg he d ble he ble c be ee ble []. The cuve couced fo he buled d c be ee fgue [] d fgue []. []
13 Tble Coeffce fo d b vou. b Fgue. Cuve fo veu. Fgue. Cuve fo b veu. Coeuely [] Now code d hu: [] ce y he ullyg boh de [] wh [] yeld
14 [] ullyg boh de [] wh d ubug [] o he euo yeld [] ubcg euo [] wh euo [] yeld [] egg [] yeld [] o [] Fo d [] Sce [] [] Addg [] o [] yeld: [] Sug boh de of euo [] o [] d ege he e yeld: [] [] [] Subug euo [] o [] o [] d lfyg yeld [] ulg euo [] yeld [] Sce [] Subug euo [] o [] d lfyg yeld: [] Now code Poduc Idey fo hec ogeo fo d gve follow: [] Subug euo [] o [] d egg he e yeld [] o []
15 eeg he e ocedue fo e fo o d by codeg geel foulo fo of h fo b [] d he clculg fo oe e yeld: Tble Coeffce fo d b vou Fgue. Cuve fo veu. Fgue. Cuve fo b veu. A eul [] b
16 Fo d e [] d egg [] o yec fuco fo yeld: [] Fo euo [] whe gve [] Ug oduc dey fo hec ogeo yeld: [] Subug euo [] o [] d eeg he uo oo yeld: [] Slfyg d egg he euo [] yeld: [] Clculg he ohe e d lfyg fo e yeld: [] Clculg he coeffce fo he e of he euo yeld [] [] []
17 [] [] [] Eeg Poduc of Ahec Te Fo of he o c Eleey Syec Fuco.e.. Fo he Newo foul Gd d Wg [] fouled u of owe he fo of eleey yec fuco. The fuco gve follow: f [] Fcozg [] yeld f [] The eleey yec fuco of he oo e defed follow: [] [] [] []
18 . The h owe u of he oo of bove fuco gve follow: S [] Whee S [] S S [] S S S [] S S S [] S S S [] Thee euo e eeed he e of ool. I h eech foud h whe hec e volved he ool c be eeed ug he o bc eleey yec fuco o u of hec e. A heoe h bee foud d oof gve follow: Theoe. e P fo eve. [] d P fo odd. [] Poof: y codeg hec uo of e.e. d by egg we ge: [] Sce [] Subug [] o [] yeld: [] Alo [] y g oduc of o fo eve yeld:
19 P [] P [] Slfyg [] yeld: P fo eve. [] d oduc of o fo odd yeld: P [] P [] Slfyg [] yeld: P fo odd. []. Eleey Syec Fuco fo Aleg Peuo of Ahec Te Though Quve ehod. Sce Su of Powe he bc buldg bloc fo yec olyol heefoe c lwy be eeed oduc d u of yec fuco wh ol coeffce. Code e of yec fuco of by hec e follow: The eleey yec olyol of vble fo of d yec fuco e gve follow: Ode O [] The ecod ode c be clculed ug uve ehod follow: e he ecod ode be O
20 Code hec e wh d he buled d gve follow: Tble The vlue of whe _ Plog he d fo oe vlue of yeld gh Fgue. Su Su_.Su -. Su_ Su_ Su_ Su_.Su - Su_.Su -. Su Fgue Gh of veu fo oe vlue of
21 e he d Ode be follow: O Collecg he coeffce of fo oe vlue of yeld Tble. Tble The vlue of d vou vlue of. Theefoe he d Ode fo c be we follow: O Reeg he e oce fo vou yeld: Tble The vlue of coeffce wh vou vlue of.
22 Tble The vlue of coeffce wh vou vlue of. Fo he Tble c be deduced h c be we follow Theefoe he ecod ode c be we follow: d Ode O [] d Ode O [] h Ode O l l []
23 h Ode O l l [] h Ode O l l [] h Ode O l l [] h Ode O o l o l []
24 h Ode c be clculed by ug he e coeffce ued Ode h gve follow: Q Q Q Q O o l o l [] Coeffce [ ] Q Q c be clculed by ug oduc dey of hec ogeo whe he clculo gve follow: P [] P [] Cog he coeffce yeld: Q Q Q d Q clly Q []
25 The h Ode c be clculed follow: T Q Q Q Q O o l [] The Geeled Euo c be we follow: z z Q T O [] Whee coeffce Q d eve fo odd fo [] The l coeffce T c be clculed by efog le ue ehod ly o vou vlue of. e he l coeffce h fo: T [] The vlue of T vou vlue of c be clculed follow: Q O T Q T fo eve []
26 Q O T Q T fo odd [] Whee d Q coeffce gve follow: Q Fo oe vlue of we c couc he o ge he coeffce fo olvg T below: T [] e he euo vou vlue of ged fo follow: T T T T [] Whee T T T T T N d [] Theefoe N T [] Thu T N [] Thee ode e ueful o couc u of owe fo he hec ogeo he u of owe fo he hec ogeo c lwy be eeed o eleey yec olyol.
27 .. Ug ulol Theoe d Poduc Ahec Te Geeg Su of Powe fo Aby Ahec e. The ulol Theoe e h f oegve ege he K [] I h eech ooed h The ulol Theoe fo by hec ogeo c be eeed he owe of hec u decedg by fo ech ubeue e.e -. The euo gve follow: K [] Th eloh cully he buldg bloc fo u of owe of by hec ogeo. The u of owe c be clculed decly fo h eloh; howeve fo lge he clculo would be edou. Evey ool e he ulol c lo be eeed follow: & & [] Whee: f eve f odd coeffce eve fo odd fo d [] Theefoe le code he euo below fo [] Solvg he coeffce yeld
28 [] Whe he oluo gve follow: [] Clculg fo oe vlue of d he buled d gve follow: Tble The vlue of d vou vlue of. - - Theefoe fo ll he euo gve follow: [] Whe code he euo below β β [] Solvg he coeffce yeld
29 β β [] Whe he oluo gve follow: [] Clculg fo oe vlue of d he buled d gve follow: Tble The vlue of β d β vou vlue of. β β. Theefoe fo ll he euo gve follow: [] Alyg he e ocedue we ge euo follow: []
30 [] [] [] Cobg he eul fo ll yeld Fo odd [] [] O Fo eve [] [] Whee fuco of eoull ube d.e. he deoo of o zeo d f deoo zeo he eo of he e e he l fo of o fo odd d eve eecvely. Code whe
31 [] Regg he euo [] yeld [] Sce he [] [] Code whe [] Sce d he eleey yec fuco of d ode he []
32 Regg euo [] yeld [] The ulol lo c be eeed follow: [] Regg euo [] yeld [] The ooed coecue eee h fo ll ool fo by hec ogeo c lwy be eded follow: fo odd [] fo eve [] β β β fo odd [] β β β fo eve [] γ γ γ fo odd [] γ γ γ fo eve [] Collecg he coeffce yeld: γ β γ β γ β fo odd []
33 β β γ β γ γ fo eve [] Whch he fo of euo [] whe lfed.. Foulg he Geelze Euo fo Su of Powe. The coeffce volved he olyol u o c be lfed follow [] [] [] [] [] [] o [] o bol eo fo [] [] [] [] [] [] The geelze fo of c be we follow c [] c
34 O P [] whee. The olyol c be eeed P C. I ode o defy how he coeffce e foed ech e he olyol buled ble. The buled d gve he Tble. Tble The e vlue fo P. P e d e d e h e h e h e h e P - P - P - - P - - P P P C C - C - C - C - C - C -. C y bulg he vlue of C C yeld ew d d gve he Tble. Tble The e olzed vlue fo P. P e d e d e h e h e h e h e P P P P P P P C C C C C C C C C C C C. C C
35 Plog he P cuve fo oe yeld Fgue Gh of d e veu. Fo h cuve we c deduce h d _ e [] Fgue Gh of d e veu. Fo h cuve we c deduce h d _ e [] The e c be ewe T γ f whee f fuco of d γ coeffce whch deed o he eoull ube d. Alyzg γ fo oe e yeld:
36 Tble The vlue of γ d vou vlue of. γ γ ζ - Fo he Tble ely P c be fouled follow: f [] Sce e P he T [] Theefoe P gve follow:
37 P [] Whee he eoull ube d. Whe P [] Whe P []
38 Theefoe P []. The Devo of eoull Nube fo he Geelzed Euo fo Su of Powe. I ow h he geelzed euo P zeo whe. Theefoe he coeffce P c be ued o fd eoull ube. Few eoull ube clculo c be ee follow: Code P Sce P whe. [] Rewg he euo [] yeld f fo ll N [] Whe f
39 [] Whe f [] The eoull ube fo whe > c be clculed by ewg he euo []. The devo gve follow: Edg euo [] yeld:
40 f [] Rewg euo [] yeld: Sce Theefoe []
41 . Devo of Su of Powe fo Iege ug Geelzed Euo fo Su of Powe. Seg d h euo [] educe o clcl Fulhbe Su of Powe fo ege. Powe u fo Iege geelze euo gve follow: [] Sce euo [] becoe: [] Whee d eve fo odd fo Fo o []
42 Sce Euo [] educe o: []. The Vefco of Su of Powe Geel Foulo d he Devo of he Su of Powe Though ol Eo of Eleey Ahec Te. y g u of owe of eleey hec e fo o fo eve d odd yeld: Fo eve [] Fo odd [] y codeg ol eo of y whch gve follow: y y y y [] d ubug d y o boh euo bove whee odd ube yeld he: Eve
43 [] Slfyg [] yeld: [] Odd []
44 Slfyg boh euo yeld: [] Fo eve lfyg [] yeld: [] Fo odd lfyg [] yeld: []
45 I foud hough hcd Sybolc Oeo h [] Thu Fo odd [] Ad Fo odd [] Slfyg boh euo [] d [] yeld u [] whee eve fo odd fo u [] Whch decl o euo []. Cocluo. The geel euo fo Su of Powe eeed h e c be eeded o y ohe ue due o lcy d eleg foulo. Th foul clude Fulhbe u of owe d o of ohe foule deved fo u of owe becue of eeo fo of he o bc eleey yec fuco of hec ogeo. Sce ege of hec ogeo offe ew fo of u of owe leve o Fulhbe foulo. A fo h h geelzed euo c be eeded o el ube owe whch e ueful fo eedg he u of owe o Re Ze fuco d uecl ly of uo fo ol d ol owe. Whe e o he geelzed euo educe o Fe Theoe d eeg Fe Theoe olyol fo of yec fuco. Thu offeg ew gh he Fe Theoe ude. The e of e eled o h eech c be foud he efeece []-[].
46 Refeece [] Hd H. "S Hee Tgul Nube d Pyhgoe Tle." J. Rec. h [] O'Coo Joh J.; Robeo Edud F. "Abu e b uhd b l-huy Al-K" ctuo Hoy of hec chve Uvey of S Adew. [] Kz Vco A Hoy of hec: A Ioduco d ed. Redg.: Addo- Weley. [] ev.g Pooo e Hohev The A of Clculo. [] Edwd C.H. The Hocl Develoe of he Clculu New Yo: Sge. [] Ho Tho De Nue Tgulbu e de De Pogeobu Ahec h by Addol S ff-v. [] Joh Fulhbe Acde Algebæ De de culoche Iveoe zu de hoche Coe wee cou ud ofe wede. Augug bey Joh Ul-ch Schog. Cll ube QA. F f ATH Sfod Uvey be. [] Dold E. Kuh. "Joh Fulhbe d u of owe". h. Co. Aec hecl Socey :. [] Pcl le Te du Tgle Aheue P. [] eoull Job A Coed el. [] Wll Y.C. Che Ay. Fu d I F. Zhg Fulhbe heoe o owe u Dcee hec Volue Iue y Pge -. [] Ib Y. A oe o he u d d d. Ieol Joul of hecl Educo Scece d Techology. [] Guhe N. Su of he -h owe of ucceve e of hec euece: b b b - b Ieol Joul of hecl Educo Scece d Techology:. [] XNUERS. - ul Peco Flog Po Coug d Nuecl ehod fo EXCE h://dglde.lbeo./foe/de.h. [] Séoul R. "Newo-Gd Foul.". Pogg fo hec. el: Sge- Velg. -. [] Abd Shuo.S Dec A Tey of Syec Fuco P Su of Powe h://v.og/b/. [] Abd Shuo.S Dec A Tey of Syec Fuco P II Su of Powe fo Aby Ahec Pogeo fo Rel Powe-P h://v.og/b/.
47 [] Abd Shuo.S Dec A Tey of Syec Fuco P III Aoch Devg Geel Foulo fo Aleg Su of Powe fo Aby Ahec Pogeo. h://v.og/b/. [] Abd Shuo.S Dec A Tey of Syec Fuco P IV Ug Su of Powe fo Aby Ahec Pogeo o Fd Aoo fo Su of Powe fo No-Iege R-h Powe d Eeg Re' Ze Fuco Io Syec Su of Powe Fo. h://v.og/b/. [] Abd Shuo.S Dec A Tey of Syec Fuco P V Ug Su of Powe fo Aby Ahec Pogeo fo Sude of Pe Nube h Coe Wh he Euo Foed Though New Coecue of Syec Fuco Rule of Dvo h://v.og/b/.
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