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Sums of products of terms from linear recurrence sequences
Discrete Mathema!ics 28 (19791 65~79 @ Norlh-llolland Punlishing C(,mpB~y
~]lJMS O F P R O D U C T S LINEAR |~ECUNRENCE Da'~id t ,
O F TF~r~IS F R O M ~JE~)I_F r C E S
RCSSELL
U,/A {lec,,J~ed 14 At~u~l iU?H tle'ei~e!{ ~ Miltt'h 1,~7,{ (llvefl {I ~,Yl *~f ~e(lLlellt'e~ de[ll ell i~v llll~fl{te~tlrfel1~L, l~hllh*H,. , dl~q~{*l.l 1~ dm~, 'llu~]{l*l {hldJlqg lhO {fl~iel{l~ilv({~I$ld~fil*lle)~Um~ ',' prnllm'l~ ,if l,.l{fl~i{I,dtl h ~ ~,*qlH¢'tl,c % r,,'.' ,ffllphL ),}: ,. ,, I / - - Ir;,]';, ~, ~ll{~fl IP~} {~ ! It" h.qli0flt'v ,If l,{l},,I,,l*','{ ,~{{~J},~{,i fl}~ ,m"h{~,l I~, ,qqtlkM~{. uih{?l{ 111£,~(lhb/~rifa~ Ill {Jl{~ [[*['I{{F,{el l[l~ pf{,dl{Cl {*I,~iiP{m*f hPnt!i'm~ ¢,~ b ([h*~ h({f4lllhllJcPl{ vqHA, hJ,.~ The ~llrn~ fIHI!p{ l~l~e11p{~h~o¢{ 11~ il {{llVP*{~*ffHhJflPa{(ll9~,t [.,hill{,~'~l,¢~¢~,'.eh ',qm l~ ~, llmdo¢l ,ff ~;Wl*~l~, ~}eld ~{wh fil¢~l,~fi~ LI l~llll [l,){~l¢,I)~ ~,1 IIW ,{flg{l{~,i~mlhem,'~, i'lW r,..qdh ~m~r=di/L, llllt,{,hHm{lll}lioilh hml,d ill lh¢~ lil*,rla{m, ilnd l}r~,vi¢lr,,. ,,.{i,'leh~r met hllllh ~,lIv
I. Introduction ('o11%JdsJr lhe J.IHt}wJl1~ i w o JJl1~a7 r~JcLlrl'Ufi¢c
v,,, ~ =; lh~v.
With
allpropriale
+ Ih U,,
I t ....
,. It is ~iOll1¢|illl~S desirable
u,,
~ h,P.
initial condilknls
~
Io form
rehpJ~I15:
,.
he values of u. and u,. are delerndlicd stuns of lhe h llowhg
h)r all
nalul'~:
u,v,,
Iii
J~.xact expressions for the u,, and ~l,n can [1o dc~(ellXlincd f r o m !lla p l o t s I){ ~hu ~.:3rr,~spOlldxng eharltcPJl'lStic CI.JUIIIJOII~ of the IJll('ar rccurrenc¢!; illliJ f r o m the in itia l c o n d i t i o n s . Exact e×pres~ion,; for the sums t I) ¢~!!I then be dcli2rn]hled. T h e s e e x p r e s s i o n s , h o w e v e r , are wri~.ten in t e r m s o: tile roots of tile cb ~racterislic ,;,quations a n d are not, in m o s t ca: .~: i,:tuitively tl/~pealing, It ix m u c h m o r e d e s i r a b l e to exp, :ss the s u m s ( | ) as a linear c o m b i n a t i o n of ~ l i m i t e d set of t e r m s i n v o l v i n g the u, and v,, t h e m s e l v e s , 'The f o l l o w i n g ~um f r o m [1] is an examplc of the types of sums that are p : e f e r a b l e :
H~H~.2,,,®, ,,~,,
=
z 2
~H ........ -H,,,+,+H,~H2,,,,,, [ H ....... , - f I h5
2
....
if n is even if n is o d d
(2)
I~h
D,L, Ru,;sell
The 11,, a;e generalized Fibonacci numbers sz tisfying H . - H,, ~+ H,, ~ for n > 1, H~ and ,hr, arbitrary. This paper discu~,ses a method for finding simple ',inear combinations for the sums of products nf terms fr(,m linear recurrence relations. The n]cthod, whicfi applies tt~ a hlrgc class of summ~Ltions, i.s descfibe~i in detail in Section 2, The r e n a i n d c r of the paper contains a series (ff short , trzlr~lcs
2. Delinlflo,~, ,ot:~lla:l and p r o c e d , r e ~V~' ~lfe cinicerrw~.i i',i~h a |~flitr: S~t Iff ~£~Llenu(.'s tk~l:fl~d ~ly litlt,'~Ll' r'~¢tlrlcre~r' rel~fiil~n,,, "['tic IIII, Ivrn~ ~ff ".,cq ~encu t/i~ i'~ wriflun /i,, an, I satisfic~, Ihe lccurren,.'~ Ii,, ,~J~ .... ~ ~,/~,,
~ ... i .,,f,. ,,
for ;tit H.
when',: tip, / (): lhtls [i i~ a I~'Ili'I'~IIC'(~ I'~illJi()H ()~ I)~'¢k~l" I;J, Initild values Inr i~, ('OIIs(:('zlliv~ l¢:ffn ~, clnnPz¢,(cly !~p~¢'jfy the suq.cnce (/]}, i f , limit.~d nlsml~ttr ~>f r~'¢:tn'rl]rlc~'s ~n',:' being clw~,hlerud, we ,Iften writu f ,g, h . . . . i.l!~l~ifd nf t l , / , , f~ . . . . Io ~lvoid ~,!lbr~.'ript~, /~fJ Ill-It'/Ill I'!(ll, ~. b, f), wlwrc a tind b ~lr¢' III~HI3iOq ,If cOrlrjlllll illl(~CIr=V~ljtnSG L,xpleS~,il)rls (rl(:~fllor II rliv b lll'o j llrlcljllrlS of Ii ) lind f i~ ~lj~ m-ttll~le tlf SU(In~ncC*~, IS ~jnll]ly the prt~dllcl (If Ir tejlllg ~'l'()ln I}10 individnlll q~-'(itlL'HcO~:
I1( lr. ( tl I . . . . . o,. ). ( h r . . . . .
/I,,, I, ( t a. . . .
11. ))
Note lrmt tilt. ~[IbSk'ljPj (If L'flk']l ~llCi(n jr1 1110 on-lOll1] iS lI linear f,:ncthn] ,,f n. l w ~ Itl lotrnr I:lOr, a,/k [) Inld I](H, a'. b', f'! me ~l.llhlr if [ [' ~t~ r~ a'. j.~,., I¢" ;lick jlldjkj, jll;ii tel:ti~rerlce quklllellcC tlfltlPl!lloli~q Ill' file IH-fl_*tnlr aru jde~lziued al}d ![ thu ~llbrciild~, Iff the clnt~p~fiell|r dilTel (rely by I1 v~;,Iglalfi, t] I~ rjltdbll- fl~ a stiff1 td I~l (k'ilt~ if H ~g dnlibfl II, otlclt ,d rite Itj-|L~lj!:tq (~. (jt~ r~Jltt. W~ rlje iifl~H:~k'd jjl fj/tdlitj! riffl1~JO O~jl/~,'~rtOfi~ h~f rliftl~ lif tJlO feHli eUfllJi!llfl~Jl~i~
I~t |Olflt~ tltfll file ~htfli~il I(1 I]IDI. iL II ~'i~
it,ld 121 ~ : , ~ : ~ . l / l l l : l l : l l : ~ t .~1t~t :k;lk It fill: ttlt ~:!ijHifft!~k':~ tl]~lt ~illh[~ fllu It~[:lill:k:l|~k;~ [ lJk~: lllt. ,Ill Ilfitfifl ~.'t~llktilJt~tl~ Ill Iltk~ ft%:li/i~::Ik:(~,~l illt*! [tll till ~ ~-~: :1 hlt~ if) til]tlJl]~ :~:ill) ~: lillO t11~-~ Jlld,ffillJt~: .~lt:itl ~t[ 111~~ ~tlfl|l]Utillllt If IWII :ll:tt~ttli~ ill:~; ~illlJliil:: fll(~ll tllt~ #llfl!tJlIIIllt Ilfi;dlJlltl lill: ill,: tl, I{: ~'i wilh f~lll~I I t. ljl o, ll: b: [I j~ ~j¢tll~ly llt~ ttl -Ilqtl~ i h' # 1 [ t = 1 1 . 1 , . . . . . . ,/.r, Wll~-:i" l, jt~ Ijt~; ~1 llt~ l, , :: I~ Ih Ih~ll tt111, I#, b',j:l J~ il lllOOtli(:ttl I~ttlI willl r~.tl,~Cl I . I|Uh/1: h, fl: ill: lllk: ~'hil~F:: !~:llll I | 1 0 tL I t J ) J~ llll~J~'l~fillld: if llltly h~ IIIllJll~i:l If #v/~ y I11111-~ffll II:lftl ~lf il ~l~l!d~li~l ~11111~111) h ~I ¢iItt1111J~11 t~Fttl W.F.|, /Jill 11://, 1)
Lu~e~r wruwen~,' ,,,em,'~ then X(n)
is a ((~m'~nk'cll ~um. d e n o t e d ( ' t ,
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I. Tlle~e ¢.~ists (I shlnd~rd ~unl X(II
67
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,~imill~r m Bfn. a b. f), ~uch
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if ~.'nd ~,~ly if there e~.isi~ ~l sum ('(ll. ~1. b. ! I I'!lnI,ul nl with rl"q,.~ l In PHn. d. b. t') MII'tl It (H lfln, lt, b , ( I
("ly./Lb.()
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I~ttlO|, Ally I'IIII(IIIJCII!SHill i~ it ~lIlll(l;II'dSlIIll,~llppIISC th;ll ~I '.[;lll(I;llll~,[Ilflv'~I~l'~ llIIll S~llJ~[i~J~ I}IC ¢IIIIIllJly I~III tllill ih flOt II CilIlI111]C'illS~If}~. l(nch It'llll i~[ Ilw htlilld~Ird ~lllllI!l~llJN llO1 il CilllOIlJCilIIcl'rllIlIIl~ I:~UcNIHu!e I JllI!l~I "~lll]ll]I CilHI~flJtql[ It~lllI~4l~/ SVSII]IIIIIIi(-IIIIVllSJll~ lhe ilppf(tjllJilfc I'(?CLIIICIICI'IcJ;tlJ(~Is TIlu' if|l~IIlllIlfIC~!(If I,cIIIlllll J J~ tllillwt! fl(Ll~dI~lllV gLIIIfth fill ~'HIIII/IiL'III~IIIIl-~ill l h e ~ltlerrl[!i fll find ~ , f l m d n r d SUIH fl)~ ~ II{ll, a. ~1. /') 'fIle cml< .iic~ll ~m,I I11;ly l~lll l)l* IhU "~jllllllC~l" ~IFlJ!, ~illdC' ll~lfly (¢llljv[tl('ll( gtlrtl'~ t'xjgl IIC(:ltlgC I~j' ~IIU ICCIIl IC'llCC' lehllJollg, hut If' tl gOflllJltll ¢~:iSlrl W(~ CIlI1 JJlll( ill JVHgl I~JlC relnc,,,~,fll;flhm tlll([ wilrrv ~lhlllll ghtlpJJ/iciltJ~IH itflerWHld~
Fll,.a.b.D
('ly, a.f~.f!
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I.~.h.h.
v~-t
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If !fll(l~,~1l~' If
"l]ljI~l~lllllt~IIII~i:~I~fi)lfllilijli~fjljt:ilfiHll [I!f ~mIl~jde~iiig h i l l y JndLHiIfiI~ ~,t~ i1~ n n d
I~rl,n Ihe definiti~m ¢~1' ('(m a, b, ~) il i~ ~:leur Ihllt the cenditilm o[ I1 clln(]rlic~ll ~11111 Tllt5~ ['}C Wl'ittCll ~1~
fl~r the exi~lence
('( II, ~!, b, ~') k /]1(,~ h fl, I,~ ÷ Ill, f) : C( Ih /t, t~l ' ~1, ~) fIH" ~)11 n, Ill tr21'111'~(1[ (11~! indivi.lul~[ I!l-lernl'~ the c( ndilion is
l~(n,.,!~+a,f)~.ql~(n,~l,
bFif)=~-'.c,l](I
h~a~l,f)
(4)
whele IIlo ~,mnmnlh~nn !uC I,wr I ~ I~ ~<, . . ~ 1,,,, ThIn in -. Ih~il the r,h,m ~:1" (4) t~ rl I.~lllCltlictd kufll w.: , t H!th a, b ~ ~, f ) 'I he l'.lhS, Illlly fle 1~ r!idulud (.~ee Lunllll~t I I inh) ~ canlmlcld '~':~n ~ r.l, H(n, a, h, f); the Mew e,netlleient~ ('[ of ("(m a, b, [) fill + [llll'dll(lll~ Of Ihc ('l iv ] ht!¢ [)tli'tJcllhlf I'CC[PI'I",~IiCI2 ~eqUeilCe~ t+f ~, We th~ illlett:~led 111 s~hllJt}llN lhltl 111'1: VlllJd hip all ~c[ItlcliCC~ ~;~ifi~[yilt~!lhc IL'CIIII'L'I1CL" I'ohlI]IHIN', file L'tlI1OIII'Jtll ~LIII!!i ~,(ILI~t~I d(} Illll dellelid ~ll] lhe t;lithd ~'Illl[lJlJllll~0]' ihe JlItlIVJLILIIi]I'q)rlll~l}l]I[ll~ t'eCLlrlellI:C~OqLII[I1cnm IJlldCl lhene i'l)lldili~l,~ Ihe lhlb ' I f~''~ ~tlll(illil'tll ttl--let111't WI'I ll(n, tl, h, [! fill'Ill II tim'llvlV Inlleo tllfflrlr*llt ~1'1, *I'hll'~ lhi: IIJlIW[' IHIINI'IH'IIIIlIJIIII ]CII(I~ hi I1 ~PI Of lhlO ' ' ' t~,,~ Ihl~ll[ ('qUlflhlllh: JIIL' DIII~IIOWIIN IllI* lhe ~~1L If :I NO]lll]IIll O~iNl~ lh~2 ('l d~ll2rlll]ll~ lhe In Ilu, IUillllllldl~l qfl* IIlI~ Jml~el: IJ~ ~ IillW('ll|ll'e' ~kt~'lvI1cl! ~lbltw I~ lu~plled I~l it illllHbf'l ill (iiirl( lUlL ~ hJl!ltlljllllh, ~ltllkIt ~ I'4l[ tlll~/ ~lVl'll Ill'lih;Lqll II, /1 iiil1.1 f 'LIl'r~ II*;('ll, e~dl'llut~,l ' II!I(; ~! IIIIldt' I1[ tlll~ ~hlllthllll!l nllhlllllll ~vll',t, l~llir, ,/d, b, tl) I
AI] Itl qh* ",*'fllI~.u!L'~'~ tJI I)lj ~, h4 VI(IIII IIIO ~(l'll~'l'lllbh'(t I,Ihnnnu¢t *IL~IIIIOIIL'L'h II~'~tlluLI Iw II,, t,,, I Iil, ]'Ill Illl n Shlcu' 41lll~/ hld~.dhilll~ ~illllln ,lle! l.'l)llt~h]~l'(~(l, IIlIlltll L IHIditjIHIh ItlV I]111 'g~l'~jli~ed IIIHI 11UIv hC ~Ij~ll.H'l.'llt [I)1 the' r~jllgl'Vlll nL~qOLH)¢L'~, Ill tj,i', '~L'L li()ll WI: hL~L II ( 1, I ..... i), llIId W" IL'[ ~) (h I , IP1 ..... ll,,I) hl,~ ~11 IH-I|I~llL ~ I,I Lllbj(llllY iIIIL'~ILH L'~lllnllllll~ I1i~' nhllil!L'~,l hLIIll (li thin Ivpe' in .~If,,., TIw Imnoehll~d cnl~mdeqd ,~1111~ In , ,I,, ,, I ¢'qfl ,,, ~ The 4.'¢llllljlilul Ifl' I Cliiii]il ,I I'1111 hL ~ WILjlI~.'II [111 fl~ilaw'~:
,'dl,
,, *
(q, ~ OIl, , , , ,
IIl~' hlq'~llld u'lltllllil V Cllltll~c'~; IhL' fh,~. tllt~ It L'ztlltllltu'tl] NIIM W,[.I, tle ,,. ,~illlJl.~ f,,,, hnd h, ,,*~ ~uu lili~'iuIV hlllu1"t~tllh:lll, Ihtt~ I~llll~ Iq tilL' folh~wjng ~/N|L!lli ,)f lilt~ulr I'(~l;iltJllllh:
/ ~J Ii~l,
~,~ ~ ,',,
~ll i i , ) . ) In
{I I
(~.'lll~ll~i~H
[o1' I~,,,),
~equltfl~m
fm' f,,,, ,1),
ilk Ih!lh'ite
i'{11'111 thin
Mlnlllllllk)11
b0C()llle~
(f(~l' Hl'~ollnccJ
number!+ w h e t ( I'~+=b. t ; 1 = 1 ) ~ j ~ , ~ , , t ~ ~, ~ 1 ~lnd (fl~r I_uc'~l~+ numher,, w h e r e L,, = 2, lq I~ '+-,~.~+=,, Li, = L,,~., 3, hut,i e l l - k n o w n idel!liti,u~ 111 !dlnrthand IltT~l~lioll tile ~%llllllTliltlell ~s ~l+ltl ' I i~S I'olh~wn (0) = ((1) k ~ I) = (2)
('+)
~OW ['oH,~Ldel+ IF;~ ~ullllll[l|J(ln r'~f tl ]-((~,'111 ~ / . Ir,L':,+~ ' whLIL' f ~111(] ~ Illlly 110 dJ~erelzl ~enel'~lizell I;JbOllllck'i s~,.mem'em T h e Cthl(]rlJclt] NtlTll lo I)C [o1111d IS
hl shorlhtltld ht~M++:
IcllllS wl~ Wlllll l(~ lh1(l iq.~, ~,pl. Inr~, I,a) gtl':ll lhlll lhl] Iclli~,~;ill++!
).~ (fill
: t+u+,(tl(l14 cu;(()l ) ~ on+:( lilt I cn ,( I 1 )
T h e t.'imtlllJun
It) ~lllJt+l'y (L~2111111!I 2I iN llhln lilt,* fifth WJl1~,
(~,.,(llllI
~ i,+,i(()I I I I'm(I()) + ~i ,(llI
-+ r,,,+< II~ ~ ,,,dI21
+ I',,,I21)
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+ (l'm l i'nlI(OI)
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?~ +il,11ill:ll, hi lht' l'nlh~'+vJ+l~ ny~+l++~i,l of +hill lh11.qll t~L|lllltJt)lin (l)llt] hll ullL'h tit' lhu IhlL!llily 111+.lel)elt+.ll~lll illl+ltthul'N tl+' ~(()()1, ((]l), (14t), I I a))) ~J~.'+'++ ~1 htllLIlillll hi Ihi' ~Iri~ i lfl '+tlri~llllltlnll ~' ~,, ,,N,, ,+', (L rn n cl em
<'i~
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r+,, + +'l,,
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lilt! titll llllitioh
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(fl~
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2 ~, f,,,,r ....
f,,,,~
.....
,
+
1;...... ~. . . . .
l'~t'cilll ll1+tl lh,~r+e sunltllll|ittllm tlrg {ncliH|flite Ntlllilttiilillllm Whell tile Ii111i1++I l l . tl+it+tl I(:, ewflmlh' lhg dellnlte l,lllllIH1ttlJtlnt4. I k I, (2) ltllll t+tller [tlOWll !;tl111111tllJilH'+ 1!91" the ~111111O1' lll~:2-tO'fillIll'L!l,~llNyl.'tH1f++qtll++llCeP+ [l, 2, .+1,
hi
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ilmbdi,ln
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) ~ 14111i{{){lll I Ililil~{~(~}
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I'!IC |tlI[qLIC
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J'he3 l'~:[[()V/JllJ! cLmOIliCIIi SLIlll }Ills i]lll~ J'~.~n J'Ol.ll;lL]: 2~, ((I(,01 : ( 1 Ill) ~ (I01) ~(I)l I) (00(11 - ( I 111, [ii [ll()l'CconvClll(OllilI ll(l|it|iIlllIiI~sis WI'iIIO11 I11 lh¢ following wily:
(7]
~llltlftllllh}11 ~,llll','~ll~llt ill
Ill II 1, l~'li'l m l;,I (7)
h:lq h~']vll tti!ltgJil~',r
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I,
I. . . . .
lt(~llilf: ~l{tl ' fi~l:il~ !m i I~iiltl~:l!ii41 !l;ltil i~ ih~: ~:uuiiitl!:~t h!lllt ~,'th t,~ , , ,, , itti!', !tl1 II!~ ill1~;ffc:litill~ hi 111~; m~t!lll~tlt:~]~ t1,I ~t!l!l IIF
I'
!i,
.....
'"l
..... i>,,,
'f ............
. .i...I;i
c:l~:llt/mn;
,
dtl
Ih~ ~ rilk'ill p!linl i~ Ih~li !~.q. (Hi hlih~ Wh¢li ttil 7 7i:(tlil!ll((~ Ill ~!ti~,:Vi(,~ Ih~ 51111t1~ i~L'ill'i~i!Vn p:l~ili(in ~1~ ll) l t(,r I1] It i~ ,lth~l lilllScl f(~r 111 ~r IL } TIle ~>qi¢l s'~t!llCliC'~ i1$¢11 i~ Cllll:lllL:ll~ly il'rL~il~V~lllll Iht Ii, i 171~1~~ h~ "icwl~d a~ llflrillildfdrs I(1 lilt] uqiul;iliil, ~llhil~cl ilnly !o ..~li~fYin!i I ~o ~ltll~r(ll~i~ilc rccliri~ilL'U~. Wiih ihi,~ iil illill(l, <~illC~ I111 illlii ULI <~iilisfv lh~ Sllllli~ I'l~l:llri(;llt'¢ illll.I fl i ril,, 1~< ill~ly hi~ ~lih.slillll{~ll 7(lr i~ ill !h~ fflrluli/ uxi;ic,~siilll (~) iln{l vic~ ;,t:l<~il:
......... , , ' " + " ' 1 ' , ,
~(', ,, ,, > ' " l ~ , , , , , . , . ~ " , f ~
(Ill lht~ l}lhor li~ind, c(lil~iil~l ' h(iw cxchlllll~cl hi cvl]ry lt~l'lll:
~'..
,".......... I"'
" l ' , ..........
(14) i~ c'll~i,l~tl
wlll~ll
', , ~ ~iild ~
~ic ~iilll)l ~,
,
t,', . . . . . . . . , " ' f ~ , , , , , , ~ , , . ~ " , f ,
......... , , , , ' "
~ ",1:',
,~ ,.
the' co~lllcier,, ~, ' ' #'1 ,tr,I ' li,' ri ' Illllhl hc idunlictl]; lhilS ('l i, :('~ ,, .i, i. '
'
72
D,L. RusseU
A s a conscqu,.:nce of I , e m m a 3 we can often r e d u c e the numb~'r of e q u a t i o n s in t h e l i n e a r ss!~tems w h o s e s o l u t i o n s d e t e r m i n e the coefficients '~f the c a n o n i c a l s u m s . W e say that (i; . . . . .
i~ ~ , i i, ii~l . . . . .
i~ n, ik, i~,~ . . . .
provicl,ed tilat {j~]- n o d ~,/~} satisfy the s a m e r e c n r r e n u , relalii~n ~lnd tha~ a I = Us, T h e r(rlaii(~n -- i~:
II)(l(I] ~ (ll(l(i), J Illll 1 (IllI~) ~ (ill,i) ~ qllii!), {Ill) I (lltt~+(l(llit
fill]
I)lll,
(ill)
I ' h e i~tlllt.etl datl~l~lik'lil ~L~!I1 i~
~',..,llll'Ill
~ ~'l,,,,I lll[li + u, ll,,j I Ill] I n,,,,j !l I I,
IW~< tt!'ilO iv I hl~toiitl ill' I' I '~¥htql llSill~ redllee(| l!illlilllit'lll terror ) I ql! !tit' ~tlnnlltltltlll ill lilt' .1-11~rlll tile l?'qlfltllllllll i,f J.t'nlnlll 2 t'llll fie t!xl'li'l..ssetl hi I?1ii1~ ill > it'd!it i,d L'ltllinlit!tl[ lel'illS lit hillowt; w,,,,,,lOlllll ~ tv,,.,I Ill1114 u'~.,[I Illl q W~d
II] b[I Ill -
-w,,~.l I I I I ~ w.,<,l ~ ill i u't,d2211 ~ ~,,,H,r2221 !he hliililllil~i [111 I'halll~Jll~ Ihe I,h.a, I~ t.'linllllll:tl' fltrrll mum illlil hi! Wllilen hi It, tlllU'i:tl t'illlllllit'iiJ liir!ll',
I ! i l t - Jlll!l~llllll, I.'--~il I;~STI
3111,i~ ~-Illill÷llll~lt. I I I II ~1 1 iUl ~ I lll~ll ~ltltllll
II,c it'~illlhi~ iil~llli~ Idtlll~ilillil t~l Iho I'li!ilitt'lll~i
II
I
I
IJ
IVlt"j-
l
; hleorrech e, ~e~e~lz,erwes
73
The solution is w ~ , n . - w , l l = ! w~,,,, ~ ') .~;m=~. which i~, consistent with Eq. ('~). In reduced canonic~d term the! ~t~lt :on is written
2 )~ (ooul-
[! Iol -[ono]
[l~l
The system of }il]ear equatinns has been reduced from a .iystem of eight equations to a system of only four eqaaii,3ns. (~s irlore [actors are included ill the ~ttmlnations (m increases) the reduce'on if: ':izc is mnre dlmnatic. The shorlhand notation for the ~t]mmatkm ~lf H~e 4-term )f (,,,~,, ,il.,,k ..... it; as follows (in reduced eanoo'~:td terrns): ~;~.[e~)Ulfi= w,,,r,,.fOnlOl~ w.,.,,[ I(ll)0I I wH,,,,I 110()] :w,,mllIH)l+wm
[lllll,
The: eoildifilln ll~ t~e satisfied I] t2tllllla 2) ~i IhL' !'(i,l~)Wilig;
t~',,,..,,[OlltlOT t wm,~,l If)fll)l t ~ u,, [I l()lt} I w~qt,,[I 110l~ v.h~,[ I ti ij, 111111
w,,~,~l I I I I I ~ tv =,,~,121 I I I t" w~ .,.,~122 e I 1 w ~ , . i 2 2 2 1 1 ~ w,t H122221, The fllilowhl~J fol'intilt!~ Itl'e u~et h~ c!1t111~ctht' I',h,N, t(~ i'e~,hlt'etl f'Hlllllliq!bl Itnrll; 121111
,l[1111t~ I l l Ui],
1 2 2 1 1 1 - ~ 1 1 1 1 t l ~ 311. In I i 1101)], J22211-,I[1111]~311!10]
2[linOl~ImI)t)],
T~le I'Og~.llllll~IlllelU ~3"glL'lll t'4 ~hi)Wll I~eh)w:
lht~rol:~
l)
I
|1
- I
wm, lu
l)
II
{I
2
I I a/ll""
U
I
3
2
I
Wi~m
II ~
[I
(t~)
I1
lift ,~tfltliltlf~l ~11t11} t~(til ~l.~l~'i ~filf [ll~ ~Jtttl tA¢ I~IP 4:l~ffll
~ (i ~11;i)}: (:I'H n~E~
lit.. tC.,ss(ql
74 solve the flfllowing equation:
~. t.,,.,A 0cttl01 ~ .,,,.,,110(10] ~ .,, .,.[ 111)()] - Ut~n.[lll0i+n~i[II/tl] = w,,,..[0tMl)] + w.,r.ll()0(}] t wH,,,[11(/ 4 w,l~,,[ I I ! l i t + wnnll[I I I 1 ]. W e will Ionk for n - n J '¢' such that the above equatk:,n holds. By act c~bvious extension to Lemuna 2 ,~rly .~oh|tioa must satisfy the following condition: w,,¢,.,[(1000' ,~ ~ ~:..[ | (100] + w: t..[ l 1 Off] + w~ ~~,)1 ; ] !0] + w ~ ~[ i I [ 11 ~ n,..,,,[I 111 + n.,.,,[2111 ] + n, ,,.~[22 ! | ] ; ~h, .~,F2221 ] + n H ~, [2222] = - w . . . o l l I I I ] + w.,...[21111 '~ w ~ . , . [ 2 2 1 1 ]
(1 I}
+ w~ ~.,{22211 + writ ~[2222~. It is mos~ c o r w e n i e m to change this equat i on into canon~ca? form with respect to (4 l t I) (instead of w,r,t. (00(i0) as earl i er in this secti, m): [0fl00]-[2227,]-[222l]+[2211]-[2111]+[1.11], [1000]-[2221]-212211]+312111]-411111], (12)
[110i)] = [ 2 2 1 1 ] - 312111]~- 6 [ 1 1 1 1 ] . [I 1 lift
[21113-411111].
T h e rc~ultin~ ~latrix eqt2ation is ~hown below:
2
(}
I
0
- I
(I
()
0
0
I)
wHo~ = -
0
w~t~,,
njw.
.
(13)
trio
-iJLwH,,j
L,Z,,,, J
The illatr'ix is ol rank ./. and we conclude ~flat solutio.., exist, provi ded that Illllll)l ) + ]~ 11 [ {](]ll
tl i VI¢~
2nlll(,+//11
i i = ()
(e~sily derived a:Cler finding the row-echelon form of the matrix [4, pp, 4 8 - 6 2 ] ) , ,'Some of Ifle pos!;ible SUlllroatiolls are the follov.'ing:
31~' {I lontli4
[ I I tilt}-- 211100J4 311 110].
3~!~ {ll0,~lll] 2JOllflll]} -]JltlIlJ, '~ I[lOcllfl- II Ill01 *ill IOl I l I I l l}--[I)O00].
(14)
()ll ihl} eel't hmld side of these equatiollS the stlnllilallds ¢(mt~iSt of I'edtlced c/ttl
nluJlipl¢ canonical terms |llld¢I the ~,,n,.l~:atitn ot~[ not reduced t~lllllrlic[l] (trills) the r.h,s, Ilecd not 12OltSi~[i?[ I'clhlced qllnonil ,I [el'IllS, A silnp[c eMllll[llc i'~ Ihq~ folh'Jw hlg sulnl'Aation:
',~ t( t l(I I l f (O{[ul) ~ (t)l ~ I ) ~ (() 1()0)} : -(()llO)+(()illl+(luO]~
~(1101)
:15J
This summation ~and many others) may be derived by fifth,wing the steps '~f l~q~ (1(1), (111, (I 2) and (135 for tile full system of lb cammical I, rms. The re'm~finl, | 6 x 16 matrix equation (tim analog of Eq. f13), is of rt 'lk t(); E q (15) is just ,~llt: of the p.'/ssible sum nail(ms, Note the close relatio~ ~)1'Eq. (15) and tile first of Eq. ( 14): if all permutaticms of the canonical term!; :if Eq. (15) arc added tot!ether, tilt: first equation of Eq. (14} is obtained. Remember th::; each canonical term in these sums represents the product ¢)f terms from fore distinct sequences (aii satisl}ing the generalized Fiboni~cei recurrence relation' If only one sequence is involved, then some ,ff the equation!~ may be simplified somewhat. Equation (15) becomes the followinR, for instance (in conventi, mal notation):
E (f?d,,,, - f,,f2,,5: :,j:,,, f,. . . . It rema?ns true, however, that tit) cahonicaI s u m exists for ~ f:,
4. Generalized ~ibonacel sequences ' "ontinued) In this section we relax the constraint that a - ( 1 . . . . 1) and allow a to be an m-tuple of arbitrary integer constants a - ( a I, a 2. . . . . a,.5. As in the previeus sectiuu, all sequences {f,} are generalized Fibonaccl scqueno,s defined by f,,, = (,,,, ~+(,.,, 2 £or all n and b - ( b ~ . . . . . b,.,) is an nl-tup~e ol arbitrary integ¢l constants. Consider the slim of the I-term f2,,,,,. We assume thai
Y h,,,,, : ,,,f ...... + ~:f . . . . . . . . and derive the ¢c,ll,~wing condition in horthand notation (Lemma 2): c~;(O) + e~(I)
F (2) - (',,(25 + e,(:!L
Using the relations ( 3 ) = (0)+ 2(1) and (25= (()5+ (1) wc find tile r(~llowbll~ systcnl of cquation~:
It, [
[:'ll:
tile solution is ~,, fl. c I ]'~ '['l
Ill,
I. '['herefm'c we have the frith*wing sllltllttnli(n.:
%
tll
l'he s~me prn~pdm~ Iolh*wh~g forlntlh~h:
I~lmo¢l
~pplletl In lhe .c~ps wh, '
-(3), ~4),.., leaO~ to lhe
2 ~ t~,,,,, ~ f ...... ~ f ....... I = f .........
(16)
and ill general
(Lk
1 ( i)"}~'.f~,,,,,-(bi,
, - ( - 1 ) " ) f k . + , . + F ~ f , ...... ,
= (~,(,,, ,,,,, - ( - l)"fa ...... where { L j is the sequence of Locas nmnbers (L~.- L~ t~ Lk z, /-.o-2, L~ = 1) ard {F~,} is the sequence of kibonacci numbers (Fk =F~ t+F~ ~. F ~ = 0 , FI = 1). The ]asl ..quation of ( t 6 ) u~,;es the well-known identity [~,+~-.F'~ ]f,,~+F~[,.,, which is easily verified. Com;idcr the sum r.f,,.,g_.,,..; the canonical terms in shorthand notation are ([tl]h (01), (10). and (11), The condition to be satisfied is the following:
c(,.(O0)+cm(Oi)+em(lO)+eu(ll)+(12)
=
- ca,(12)+ cm(]3) + cm(22)+ cu(23). After changing tile r.h.s, to canonical terms and solving the associated lineal S}'tcln. the tanonica] SUm is found: 2/.V !t)0) = ( I ) l ) - ( 1 0 ) + (1 l), (171
or
2 2 f,,~,g ..... = f . , . g . . . . . . . - f . ~ , g z , , ~
+ f.-,,tg .......
tThis ~am may be simplified somewhat.) TI',e same teehni~ .'s lead to tl-,e conclusions that the sums ~ fz.+.g2.+, and ~ L,~,, ~.h~,,,, caanot be expressed as canonical sums. In both cases the matriee... of the assodated system of linear equations are singular matrices and no solutions exist t,.) the systeras.
5. General recun'ence sequences
The methods of this paper can be applied to a wide variety of situations. Consider the Kzneral second order recurrence sequences {r.} and {s.} defined by
r,,=ar,, t+br. 2 and
s,~=es.-~+ds. 2.
The cc, nd~lion of Lemma 2 can be reduced to canonical form with the following
((2)=~(111~ #(!(1),
(21t--~(ll)+h((ll), (22) = f,c(l li F ~hl(]',~)+
bc(Ol)+fid((lO).
T~le restllth~g linear eqtlation is a~; i'oiinws:
11 I
1 -d ,-
b 1 -n
h( b',,,t = [il)]. / r -mll|ll~,,t !,-ac~j,:,,J
(i8)
,
If a, h. c, and d are given, the solution to (18) defines a canonical sum for r~,+~s.,.. For enan~ple, if
r,,=4r. ~+r,, ~
s.=-Ss,, tks,, 2
and
the resulting summati(.~n is ~ r.s. - - r . ~is~ - r,,s~,+ i,
If a - O , b
=
~r,,s,,
! and
c: ( d -
1)2;~ 0 the retulting s u m m a t i o n is
1
c2 (d_l)z[d(l --d)r.s.+cr.s..~+cdr.÷~s.+(1
d)r,,,~s,,.~]. [
Ug)
Initial conditions of ro = t. r ~ - - 3 give the s u m ~ . . . . . . ~ ( - l ) " s . ; akernatively, initial conditions of ro = 1, r~ = 0 give the sum ~ z i ~ s~i. T h e same m e t h o d s also apply to h i g h e r - o r d e r recurrences. As a final e x a m p P , let {t.} be the sequence of generalized Tribonacci n u m b e r s defined by t. = ~._~ ~-t. : 4 ~-3. It is easy to derive the following ~um: 2 ~ (0) = ( 0 ) + ( 2 ) or 2 ~ t,,+p = t.+. + in+p+ 2.
6. Criteria for non existence o| the s~a~lard stun Several of the s u m m a t i o n s conside;ed in this p a p e r could not fie ex ,rrssed as s t a n d a r d sums. T h e reason for the non-existence of standard sums ca : Oe better u n d e r s t o o d if the solutions to the individual recurrences are wrilten u terms uf the roots of their characteristic polynomials. Let a~ oe any root of the c Faractedstic polynomial of the sequence {f,} and ~ssume that a = ( l , 1 . . . . . t). T h e n suppose for Ihc m o m e n t that all of the roots are ~,i~tinc| and consider tire expansion of tile m - t e r m B(n,a,b,j). O n e of the terms in the s u m m a t i o n is
lit~ illwil%, lllie~ hld~lilflI~ hqlilltliflJiqlIttH,Sl lh~ r~'.,si ~,~ IIfl lh~ lJlllJl~ fl - ~/ ~tltll
II
~.
II .41111pll~: Ihld ~-'ltellJFJ~llts Vl ~ J M ~tl~qt ,
IIIL~ llllhlwJllB CllltdJlilllt is
l']leI: iiilch ti~rIll llf i]!~J Stlllllllilliilllo! the nl-teFnl Cltll lle iedut't:d Ill it Iiae~ll' Clnlqqilation of prildu!:Is ~I' rilOls, hl tinll Jh~} SHill ill these pro'JtlClS n[ roots can h~2 [ilChll'ed inhl p r o t h i . l s ill tile 1"12CnlTcnk'~gt e r m s ; the resulI is ~ite l'antnliciil SHill. It mnlliple r n m s esisI o r if a .~ {1, I . . . . ,I, Ihe p r o c e d u r e is s l i g h t 1 ) d i f f e r e n t , hut tile e n d resuh is the Sltllle, T h e m e t h o d o l o g y described hi Sel tkm 2 and illustrated ill Suclions ;! 5 is essclltially a wily to d e t e r m i n e tile cneilicien~.s q wJihollI havin~ Ill lhld Ihe r~lllts of tile cnilrilClelisiJc eqtlati(ms, [l ~ ¢ t e . . e~,,, - 1 Ior s o m e set of foists ~ , , . . . , ~,,,, the slalldard s a m doe!i I1~.~1 exist, hi lh[ll cilse the.: ~llqerm tO be sal?lnlCd contain~i a t l l m Of t h e form ' m,,]' = ~ c( I t = on,
V
which ca!t~lol he I'acR~red into a p r o d u c t i~f r e c u r r e n c e ternls. This is illustrated in m o r e deill]I fol s e c o n d - o r d e r r e c u r r e n c e s i i {51. SittialJolls encounh.'red hi this p a p e r w h e r e no s t a n d a r d s u m exiats can now he.' e×phfined ~ e t If,,}, {g,,}, {h,,~,, {k,,} be F:bonacci s e q u e n c e s ; t h e c h a r a c t e r i s t i c eqtl~tlI~lll is ~2 -X-- 1 = 0 and the roots are
~:,~ll
~/?=
1.61g,.,
and
,~=~(1-.].~)=-0,618
... .
T h e sun:.; ~_~[,,g.h.k,, ~ (.,,g.~,,, and ~ , , g , , h 2 . all o w e t h e i r n o n - e x i s t e n c e es stand~lrd sums to the fact that & : ~ 2 _ 1. In t h e case cff E q . (19), t h e c h a r a c t e r i s t i c polylvmfial of {r,,} is v 2 - 1 = 0 w h o s e r o o t s are 1 and - 1. T h e s u m m a t i o n does not exist as a s t a n d a r d sbm if t h e c h a r a c t e r i s t i c p o l y n o m i a l x e - ex - d = O ok Is,,} has a root that is e i t h e r I e r - 1. If c -- d - I = 0 t h e r o o t s are c - 1 a n d 1 ; if ? - d + 1 = 0 l h e roots are c ~ 1 und - 1 . T h e s e a r e precisely the cases in w h i c h (19) does not hold N o t e that it i! crit(cal that a s t a n d a r d s u m be valid for all s e q u e n c e s satisfying tl~e n-ecurrence e l a l i o n . Oeherwise t h e initia| c o n d i t i o n s could i,e such t h a t t h e t r o u b l e s o m e t e r m de.es not a p p e a r in the nl-'~erm b e i n g s u m m e d . For e x a m p l e , let {.-I, f,-d,, f,,-L ,+[. ,.Then 4,, ~4 f.:&" and ~ f , ~ : ~ & a " = ~ = c f a + ,.
T}I0 hld~lJllJl~ ~HilIltl!llh)fl I~ ~lt~)ihlr lu lh¢ ql mt~id, ~lll IhJq J~ HoI (ill iflM)lil(-:o ul l) t'ttltltllJtJ)ll ~llltl IleQfl')~,~ jl j~ lift( v01hl fill: qli o IIClW~I htt!ihl:Vilt~, ti~ -- 1], I ~ ~:
(IIWHI It ei~l IH" ~OIJltdl!L'~ d~JILIrlliHed hy lih,~it[le; IlllelIC~ HJh!i(III~ [hi~ I):LII;~I" V~lllhll'~:s4 nlelhc~di~hlgy for ~hlding III¢ indelillih.i(fllldd~qhfilei htllIl~eli'pr,),lucl~, (i[ telIllSIfIllltlJl~ ~e'qtli:ll¢~ YlP~ ~tllt~s~(l [¢iiiiIiiil[e dihlillclJvc illseveI~II Wil~,: lit IIlny eo6~jsl o[ it lJq~aF L'onlhJilqlJiln ~[' IeIIllh, called c~IIl(IIlJt'iIIleIlI]~i (•) eiICJlcIIIlllllJ¢[l[t,atm ctlll~i~it~(i[ d pO)dllCl ol fib Itllh,iml. l~it+h)l[r(1111~+'+i:ll,ff lhI.~ origin~d ~;equellCeS; (3) Ihe ~uhscrJpIs ()l" Ihe filCtOl'S at,' lillUar tuiwIhlllS eli Ih¢ ~Hinnla~hm Wlliahle, (4} tile [dentifie~. (ff Ihe original heqnences are nlainhlinucd in the final lind:hi ~'iml[lJllal Joll ; alld (5) the SLIIllS ill'(3 valid for all SCqLh~'IlCeS hatisfyinB the FectlFCeIlCC ,-ela!J(llls. The rnetltod finds Stlch a IJllear co:t]tlJllatk)n wfienever one i.xJsts and ;llay I,e SgllllllarJzed LI!I fol) )WS: ,vrite the e q u a t i o n of the indue!ion 'step ([.¢Fiuri~l 2): (2i find a se) of canonical ~erm~ and write the c,~,,mions of ;tep ( I ) Jq terms of lfies ! canonical telms (ill some ease~ red:+ced canor;ical ~cl-i-ii~ may ft.' used): (3) write the ass ) c i a t e d system of linear e q u a t i o n s (this ~yslem must exist since t h e canonic~d t e r m s a r e linearly i n d e p e n d e n t ) ; (4) solve the linear e q u a t i o n s - - t h e system!: of linear e q u a t i . m s has a '~olution if a n d only if l h e d e s i r e d indefinite sun, exists as a linear c o m b i n a t i o n of canonic.M t e r m s ; and (5) simplify as desk'ed. (i;
8. References [1] G Berzsenyi, Sums cff wnducts of generali:'ed Fihonacci numbers. Fiblmacci Oua:l 13 (1975) 343-344 [21 J,C. Pond, Generalized Fihonacci surnmafi,~ns, Fibonacd Quart, 6 ([9681 97 II)8 [3] Dr_ Russell, Notes on sums of produch of genelali/ed Fihoqacci numbers. Fibor ,cci Quart. to appea~, [4] E,D Nering, Linear Algebra and Matrix 1 ory (J, Wiley, New YtJrk. NY, IC)a3L [:~] D,L. Russell, Summation of second-order ,,'currence terms and their squares. Fiboneco Ouart, Io appear
~]lJMS O F P R O D U C T S LINEAR |~ECUNRENCE Da'~id t ,
O F TF~r~IS F R O M ~JE~)I_F r C E S
RCSSELL
U,/A {lec,,J~ed 14 At~u~l iU?H tle'ei~e!{ ~ Miltt'h 1,~7,{ (llvefl {I ~,Yl *~f ~e(lLlellt'e~ de[ll ell i~v llll~fl{te~tlrfel1~L, l~hllh*H,. , dl~q~{*l.l 1~ dm~, 'llu~]{l*l {hldJlqg lhO {fl~iel{l~ilv({~I$ld~fil*lle)~Um~ ',' prnllm'l~ ,if l,.l{fl~i{I,dtl h ~ ~,*qlH¢'tl,c % r,,'.' ,ffllphL ),}: ,. ,, I / - - Ir;,]';, ~, ~ll{~fl IP~} {~ ! It" h.qli0flt'v ,If l,{l},,I,,l*','{ ,~{{~J},~{,i fl}~ ,m"h{~,l I~, ,qqtlkM~{. uih{?l{ 111£,~(lhb/~rifa~ Ill {Jl{~ [[*['I{{F,{el l[l~ pf{,dl{Cl {*I,~iiP{m*f hPnt!i'm~ ¢,~ b ([h*~ h({f4lllhllJcPl{ vqHA, hJ,.~ The ~llrn~ fIHI!p{ l~l~e11p{~h~o¢{ 11~ il {{llVP*{~*ffHhJflPa{(ll9~,t [.,hill{,~'~l,¢~¢~,'.eh ',qm l~ ~, llmdo¢l ,ff ~;Wl*~l~, ~}eld ~{wh fil¢~l,~fi~ LI l~llll [l,){~l¢,I)~ ~,1 IIW ,{flg{l{~,i~mlhem,'~, i'lW r,..qdh ~m~r=di/L, llllt,{,hHm{lll}lioilh hml,d ill lh¢~ lil*,rla{m, ilnd l}r~,vi¢lr,,. ,,.{i,'leh~r met hllllh ~,lIv
I. Introduction ('o11%JdsJr lhe J.IHt}wJl1~ i w o JJl1~a7 r~JcLlrl'Ufi¢c
v,,, ~ =; lh~v.
With
allpropriale
+ Ih U,,
I t ....
,. It is ~iOll1¢|illl~S desirable
u,,
~ h,P.
initial condilknls
~
Io form
rehpJ~I15:
,.
he values of u. and u,. are delerndlicd stuns of lhe h llowhg
h)r all
nalul'~:
u,v,,
Iii
J~.xact expressions for the u,, and ~l,n can [1o dc~(ellXlincd f r o m !lla p l o t s I){ ~hu ~.:3rr,~spOlldxng eharltcPJl'lStic CI.JUIIIJOII~ of the IJll('ar rccurrenc¢!; illliJ f r o m the in itia l c o n d i t i o n s . Exact e×pres~ion,; for the sums t I) ¢~!!I then be dcli2rn]hled. T h e s e e x p r e s s i o n s , h o w e v e r , are wri~.ten in t e r m s o: tile roots of tile cb ~racterislic ,;,quations a n d are not, in m o s t ca: .~: i,:tuitively tl/~pealing, It ix m u c h m o r e d e s i r a b l e to exp, :ss the s u m s ( | ) as a linear c o m b i n a t i o n of ~ l i m i t e d set of t e r m s i n v o l v i n g the u, and v,, t h e m s e l v e s , 'The f o l l o w i n g ~um f r o m [1] is an examplc of the types of sums that are p : e f e r a b l e :
H~H~.2,,,®, ,,~,,
=
z 2
~H ........ -H,,,+,+H,~H2,,,,,, [ H ....... , - f I h5
2
....
if n is even if n is o d d
(2)
I~h
D,L, Ru,;sell
The 11,, a;e generalized Fibonacci numbers sz tisfying H . - H,, ~+ H,, ~ for n > 1, H~ and ,hr, arbitrary. This paper discu~,ses a method for finding simple ',inear combinations for the sums of products nf terms fr(,m linear recurrence relations. The n]cthod, whicfi applies tt~ a hlrgc class of summ~Ltions, i.s descfibe~i in detail in Section 2, The r e n a i n d c r of the paper contains a series (ff short , trzlr~lcs
2. Delinlflo,~, ,ot:~lla:l and p r o c e d , r e ~V~' ~lfe cinicerrw~.i i',i~h a |~flitr: S~t Iff ~£~Llenu(.'s tk~l:fl~d ~ly litlt,'~Ll' r'~¢tlrlcre~r' rel~fiil~n,,, "['tic IIII, Ivrn~ ~ff ".,cq ~encu t/i~ i'~ wriflun /i,, an, I satisfic~, Ihe lccurren,.'~ Ii,, ,~J~ .... ~ ~,/~,,
~ ... i .,,f,. ,,
for ;tit H.
when',: tip, / (): lhtls [i i~ a I~'Ili'I'~IIC'(~ I'~illJi()H ()~ I)~'¢k~l" I;J, Initild values Inr i~, ('OIIs(:('zlliv~ l¢:ffn ~, clnnPz¢,(cly !~p~¢'jfy the suq.cnce (/]}, i f , limit.~d nlsml~ttr ~>f r~'¢:tn'rl]rlc~'s ~n',:' being clw~,hlerud, we ,Iften writu f ,g, h . . . . i.l!~l~ifd nf t l , / , , f~ . . . . Io ~lvoid ~,!lbr~.'ript~, /~fJ Ill-It'/Ill I'!(ll, ~. b, f), wlwrc a tind b ~lr¢' III~HI3iOq ,If cOrlrjlllll illl(~CIr=V~ljtnSG L,xpleS~,il)rls (rl(:~fllor II rliv b lll'o j llrlcljllrlS of Ii ) lind f i~ ~lj~ m-ttll~le tlf SU(In~ncC*~, IS ~jnll]ly the prt~dllcl (If Ir tejlllg ~'l'()ln I}10 individnlll q~-'(itlL'HcO~:
I1( lr. ( tl I . . . . . o,. ). ( h r . . . . .
/I,,, I, ( t a. . . .
11. ))
Note lrmt tilt. ~[IbSk'ljPj (If L'flk']l ~llCi(n jr1 1110 on-lOll1] iS lI linear f,:ncthn] ,,f n. l w ~ Itl lotrnr I:lOr, a,/k [) Inld I](H, a'. b', f'! me ~l.llhlr if [ [' ~t~ r~ a'. j.~,., I¢" ;lick jlldjkj, jll;ii tel:ti~rerlce quklllellcC tlfltlPl!lloli~q Ill' file IH-fl_*tnlr aru jde~lziued al}d ![ thu ~llbrciild~, Iff the clnt~p~fiell|r dilTel (rely by I1 v~;,Iglalfi, t] I~ rjltdbll- fl~ a stiff1 td I~l (k'ilt~ if H ~g dnlibfl II, otlclt ,d rite Itj-|L~lj!:tq (~. (jt~ r~Jltt. W~ rlje iifl~H:~k'd jjl fj/tdlitj! riffl1~JO O~jl/~,'~rtOfi~ h~f rliftl~ lif tJlO feHli eUfllJi!llfl~Jl~i~
I~t |Olflt~ tltfll file ~htfli~il I(1 I]IDI. iL II ~'i~
it,ld 121 ~ : , ~ : ~ . l / l l l : l l : l l : ~ t .~1t~t :k;lk It fill: ttlt ~:!ijHifft!~k':~ tl]~lt ~illh[~ fllu It~[:lill:k:l|~k;~ [ lJk~: lllt. ,Ill Ilfitfifl ~.'t~llktilJt~tl~ Ill Iltk~ ft%:li/i~::Ik:(~,~l illt*! [tll till ~ ~-~: :1 hlt~ if) til]tlJl]~ :~:ill) ~: lillO t11~-~ Jlld,ffillJt~: .~lt:itl ~t[ 111~~ ~tlfl|l]Utillllt If IWII :ll:tt~ttli~ ill:~; ~illlJliil:: fll(~ll tllt~ #llfl!tJlIIIllt Ilfi;dlJlltl lill: ill,: tl, I{: ~'i wilh f~lll~I I t. ljl o, ll: b: [I j~ ~j¢tll~ly llt~ ttl -Ilqtl~ i h' # 1 [ t = 1 1 . 1 , . . . . . . ,/.r, Wll~-:i" l, jt~ Ijt~; ~1 llt~ l, , :: I~ Ih Ih~ll tt111, I#, b',j:l J~ il lllOOtli(:ttl I~ttlI willl r~.tl,~Cl I . I|Uh/1: h, fl: ill: lllk: ~'hil~F:: !~:llll I | 1 0 tL I t J ) J~ llll~J~'l~fillld: if llltly h~ IIIllJll~i:l If #v/~ y I11111-~ffll II:lftl ~lf il ~l~l!d~li~l ~11111~111) h ~I ¢iItt1111J~11 t~Fttl W.F.|, /Jill 11://, 1)
Lu~e~r wruwen~,' ,,,em,'~ then X(n)
is a ((~m'~nk'cll ~um. d e n o t e d ( ' t ,
C ( n , a , b , fl
i,e~m~
~
q B I n . c ~.
I. Tlle~e ¢.~ists (I shlnd~rd ~unl X(II
67
/~. f~:
~ f! X(n)
,~imill~r m Bfn. a b. f), ~uch
I I1 (~ l
if ~.'nd ~,~ly if there e~.isi~ ~l sum ('(ll. ~1. b. ! I I'!lnI,ul nl with rl"q,.~ l In PHn. d. b. t') MII'tl It (H lfln, lt, b , ( I
("ly./Lb.()
I'I,x'
I.a
I~,l'}r
I~ttlO|, Ally I'IIII(IIIJCII!SHill i~ it ~lIlll(l;II'dSlIIll,~llppIISC th;ll ~I '.[;lll(I;llll~,[Ilflv'~I~l'~ llIIll S~llJ~[i~J~ I}IC ¢IIIIIllJly I~III tllill ih flOt II CilIlI111]C'illS~If}~. l(nch It'llll i~[ Ilw htlilld~Ird ~lllllI!l~llJN llO1 il CilllOIlJCilIIcl'rllIlIIl~ I:~UcNIHu!e I JllI!l~I "~lll]ll]I CilHI~flJtql[ It~lllI~4l~/ SVSII]IIIIIIi(-IIIIVllSJll~ lhe ilppf(tjllJilfc I'(?CLIIICIICI'IcJ;tlJ(~Is TIlu' if|l~IIlllIlfIC~!(If I,cIIIlllll J J~ tllillwt! fl(Ll~dI~lllV gLIIIfth fill ~'HIIII/IiL'III~IIIIl-~ill l h e ~ltlerrl[!i fll find ~ , f l m d n r d SUIH fl)~ ~ II{ll, a. ~1. /') 'fIle cml< .iic~ll ~m,I I11;ly l~lll l)l* IhU "~jllllllC~l" ~IFlJ!, ~illdC' ll~lfly (¢llljv[tl('ll( gtlrtl'~ t'xjgl IIC(:ltlgC I~j' ~IIU ICCIIl IC'llCC' lehllJollg, hut If' tl gOflllJltll ¢~:iSlrl W(~ CIlI1 JJlll( ill JVHgl I~JlC relnc,,,~,fll;flhm tlll([ wilrrv ~lhlllll ghtlpJJ/iciltJ~IH itflerWHld~
Fll,.a.b.D
('ly, a.f~.f!
~.'I~
I.~.h.h.
v~-t
I.
If !fll(l~,~1l~' If
"l]ljI~l~lllllt~IIII~i:~I~fi)lfllilijli~fjljt:ilfiHll [I!f ~mIl~jde~iiig h i l l y JndLHiIfiI~ ~,t~ i1~ n n d
I~rl,n Ihe definiti~m ¢~1' ('(m a, b, ~) il i~ ~:leur Ihllt the cenditilm o[ I1 clln(]rlic~ll ~11111 Tllt5~ ['}C Wl'ittCll ~1~
fl~r the exi~lence
('( II, ~!, b, ~') k /]1(,~ h fl, I,~ ÷ Ill, f) : C( Ih /t, t~l ' ~1, ~) fIH" ~)11 n, Ill tr21'111'~(1[ (11~! indivi.lul~[ I!l-lernl'~ the c( ndilion is
l~(n,.,!~+a,f)~.ql~(n,~l,
bFif)=~-'.c,l](I
h~a~l,f)
(4)
whele IIlo ~,mnmnlh~nn !uC I,wr I ~ I~ ~<, . . ~ 1,,,, ThIn in -. Ih~il the r,h,m ~:1" (4) t~ rl I.~lllCltlictd kufll w.: , t H!th a, b ~ ~, f ) 'I he l'.lhS, Illlly fle 1~ r!idulud (.~ee Lunllll~t I I inh) ~ canlmlcld '~':~n ~ r.l, H(n, a, h, f); the Mew e,netlleient~ ('[ of ("(m a, b, [) fill + [llll'dll(lll~ Of Ihc ('l iv ] ht!¢ [)tli'tJcllhlf I'CC[PI'I",~IiCI2 ~eqUeilCe~ t+f ~, We th~ illlett:~led 111 s~hllJt}llN lhltl 111'1: VlllJd hip all ~c[ItlcliCC~ ~;~ifi~[yilt~!lhc IL'CIIII'L'I1CL" I'ohlI]IHIN', file L'tlI1OIII'Jtll ~LIII!!i ~,(ILI~t~I d(} Illll dellelid ~ll] lhe t;lithd ~'Illl[lJlJllll~0]' ihe JlItlIVJLILIIi]I'q)rlll~l}l]I[ll~ t'eCLlrlellI:C~OqLII[I1cnm IJlldCl lhene i'l)lldili~l,~ Ihe lhlb ' I f~''~ ~tlll(illil'tll ttl--let111't WI'I ll(n, tl, h, [! fill'Ill II tim'llvlV Inlleo tllfflrlr*llt ~1'1, *I'hll'~ lhi: IIJlIW[' IHIINI'IH'IIIIlIJIIII ]CII(I~ hi I1 ~PI Of lhlO ' ' ' t~,,~ Ihl~ll[ ('qUlflhlllh: JIIL' DIII~IIOWIIN IllI* lhe ~~1L If :I NO]lll]IIll O~iNl~ lh~2 ('l d~ll2rlll]ll~ lhe In Ilu, IUillllllldl~l qfl* IIlI~ Jml~el: IJ~ ~ IillW('ll|ll'e' ~kt~'lvI1cl! ~lbltw I~ lu~plled I~l it illllHbf'l ill (iiirl( lUlL ~ hJl!ltlljllllh, ~ltllkIt ~ I'4l[ tlll~/ ~lVl'll Ill'lih;Lqll II, /1 iiil1.1 f 'LIl'r~ II*;('ll, e~dl'llut~,l ' II!I(; ~! IIIIldt' I1[ tlll~ ~hlllthllll!l nllhlllllll ~vll',t, l~llir, ,/d, b, tl) I
AI] Itl qh* ",*'fllI~.u!L'~'~ tJI I)lj ~, h4 VI(IIII IIIO ~(l'll~'l'lllbh'(t I,Ihnnnu¢t *IL~IIIIOIIL'L'h II~'~tlluLI Iw II,, t,,, I Iil, ]'Ill Illl n Shlcu' 41lll~/ hld~.dhilll~ ~illllln ,lle! l.'l)llt~h]~l'(~(l, IIlIlltll L IHIditjIHIh ItlV I]111 'g~l'~jli~ed IIIHI 11UIv hC ~Ij~ll.H'l.'llt [I)1 the' r~jllgl'Vlll nL~qOLH)¢L'~, Ill tj,i', '~L'L li()ll WI: hL~L II ( 1, I ..... i), llIId W" IL'[ ~) (h I , IP1 ..... ll,,I) hl,~ ~11 IH-I|I~llL ~ I,I Lllbj(llllY iIIIL'~ILH L'~lllnllllll~ I1i~' nhllil!L'~,l hLIIll (li thin Ivpe' in .~If,,., TIw Imnoehll~d cnl~mdeqd ,~1111~ In , ,I,, ,, I ¢'qfl ,,, ~ The 4.'¢llllljlilul Ifl' I Cliiii]il ,I I'1111 hL ~ WILjlI~.'II [111 fl~ilaw'~:
,'dl,
,, *
(q, ~ OIl, , , , ,
IIl~' hlq'~llld u'lltllllil V Cllltll~c'~; IhL' fh,~. tllt~ It L'ztlltllltu'tl] NIIM W,[.I, tle ,,. ,~illlJl.~ f,,,, hnd h, ,,*~ ~uu lili~'iuIV hlllu1"t~tllh:lll, Ihtt~ I~llll~ Iq tilL' folh~wjng ~/N|L!lli ,)f lilt~ulr I'(~l;iltJllllh:
/ ~J Ii~l,
~,~ ~ ,',,
~ll i i , ) . ) In
{I I
(~.'lll~ll~i~H
[o1' I~,,,),
~equltfl~m
fm' f,,,, ,1),
ilk Ih!lh'ite
i'{11'111 thin
Mlnlllllllk)11
b0C()llle~
(f(~l' Hl'~ollnccJ
number!+ w h e t ( I'~+=b. t ; 1 = 1 ) ~ j ~ , ~ , , t ~ ~, ~ 1 ~lnd (fl~r I_uc'~l~+ numher,, w h e r e L,, = 2, lq I~ '+-,~.~+=,, Li, = L,,~., 3, hut,i e l l - k n o w n idel!liti,u~ 111 !dlnrthand IltT~l~lioll tile ~%llllllTliltlell ~s ~l+ltl ' I i~S I'olh~wn (0) = ((1) k ~ I) = (2)
('+)
~OW ['oH,~Ldel+ IF;~ ~ullllll[l|J(ln r'~f tl ]-((~,'111 ~ / . Ir,L':,+~ ' whLIL' f ~111(] ~ Illlly 110 dJ~erelzl ~enel'~lizell I;JbOllllck'i s~,.mem'em T h e Cthl(]rlJclt] NtlTll lo I)C [o1111d IS
hl shorlhtltld ht~M++:
IcllllS wl~ Wlllll l(~ lh1(l iq.~, ~,pl. Inr~, I,a) gtl':ll lhlll lhl] Iclli~,~;ill++!
).~ (fill
: t+u+,(tl(l14 cu;(()l ) ~ on+:( lilt I cn ,( I 1 )
T h e t.'imtlllJun
It) ~lllJt+l'y (L~2111111!I 2I iN llhln lilt,* fifth WJl1~,
(~,.,(llllI
~ i,+,i(()I I I I'm(I()) + ~i ,(llI
-+ r,,,+< II~ ~ ,,,dI21
+ I',,,I21)
I Ill)
,
I ,,d~2),
Ih~! 4,h,~, J~ I'IIII11~c.' Ill ii l'!mI~l;Jk'ttl +~lllll 19tJ tl+drlt, lh~, ~'J, I~ if*' (I))
Illll~llll,
f21I - I l l ! l ~ l l l ) ,
lllld ~2~) +(il++.)~(l;!)
fllflJifl)11,(lfl)+(ll).
llllll lh!l~ fl+l !h'+' l,h,h, ~v,.' l)llll~Jll lhu 'H !nwJ ~1: +'al((ll)l
+ (l'm l i'nlI(OI)
I (C,,+ ; +I+)(I(1)
+ (r,m + <'n~ I 4'IH+ + t~)ll IL
?~ +il,11ill:ll, hi lht' l'nlh~'+vJ+l~ ny~+l++~i,l of +hill lh11.qll t~L|lllltJt)lin (l)llt] hll ullL'h tit' lhu IhlL!llily 111+.lel)elt+.ll~lll illl+ltthul'N tl+' ~(()()1, ((]l), (14t), I I a))) ~J~.'+'++ ~1 htllLIlillll hi Ihi' ~Iri~ i lfl '+tlri~llllltlnll ~' ~,, ,,N,, ,+', (L rn n cl em
<'i~
(I,
r+,, + +'l,,
,',l
.(i,
l'm
{.{i{)
<.if l
:
+,Ii I
+l'h~.' P+tlhllJ(111 J+l 1'++++ (II
I,
(I, {'n.l = ('111
!. Ill '+lll)t'Jhlllld
llt)liltJl)It
lilt! titll llllitioh
tl1,ly hi) Wl'lltun it~i l't>lh+w++: 2 ~ lOft) :: f i l l ) I (l(I), |i+ I11()1~' ~llllldill'd
(fl~
II(+!t~lton lit+ s!llll i~ wl'itk+ll
2 ~, f,,,,r ....
f,,,,~
.....
,
+
1;...... ~. . . . .
l'~t'cilll ll1+tl lh,~r+e sunltllll|ittllm tlrg {ncliH|flite Ntlllilttiilillllm Whell tile Ii111i1++I l l . tl+it+tl I(:, ewflmlh' lhg dellnlte l,lllllIH1ttlJtlnt4. I k I, (2) ltllll t+tller [tlOWll !;tl111111tllJilH'+ 1!91" the ~111111O1' lll~:2-tO'fillIll'L!l,~llNyl.'tH1f++qtll++llCeP+ [l, 2, .+1,
hi
P f
Ih~ ,dl,,Itll~llt; itflhlwm {(
ilmbdi,ln
)I
l~mt~FI
lot the I]llllllIuIh'lll
{{lllllfLI
II1: {h~ ~ ~L~lltl ~ l,,,,~,,,Jlll ,I
) ~ 14111i{{){lll I Ililil~{~(~}
{ i~,~( I I01 I ',,,d Iflll ~ ,,,~lifll IIiL' :'{liidJihlp
h; h~ ~[i}hi~t'd
~,ll~,(lllllil
t hlhlil)lll)
{I tflll111[| ~31 [~ ihtl~ ~ hu,,lilllj~
I {'1HI,{
{~ ~lfi
I{~(~)
II I ~ ,Ll{ I I IL : {' dml'iHI~
I ,hid I1~
I t Mult ~{f~ I i w,I JlJJl I {i,lOilJ
11F t ill
;
~(tlJi
ili~l IihI ~!liI~, I l:~II IIItl~,IIIIl: I~II~ lUlIl IiIIu i]Jll
IliII11 IilIII~IiiIll
iIlli~
4~I~}I
{IIHII liij]iilIIiil
IIIII:
Ii]~l If!fill IiIllI~I)IIl~IIiil {-~2.]I {IiIllll fllIIIlIIIlllIl~{IIiIiI~Iii!IIi!Iiil~I{II!!*{IilI ,lIH~ItLlnl givt-s iI ~ JI;liIillIll JhP IIii[~iJl~l~!1111111~!i{II!ill !II~:.~-{(31II!;
I'!IC |tlI[qLIC
(I
{I
I
II
~,,,~
I)
I}
(I
t)
{I
',,h,
II
I
,!
I
~I
I
('i,,~
{1
(1
I
,
c ....
J {l I I I
~1 I
I I
! I
I I
cHi~
l {I ] . l
SOIlllioll
'
{'(~{H)
('1~1 ~
(]
1 ~,,,.
('()~1 ~ ('1111 i
( ,4t[k
1}
('1 ii I ~ /'H) I ; ('~11~ I
~,
J'he3 l'~:[[()V/JllJ! cLmOIliCIIi SLIlll }Ills i]lll~ J'~.~n J'Ol.ll;lL]: 2~, ((I(,01 : ( 1 Ill) ~ (I01) ~(I)l I) (00(11 - ( I 111, [ii [ll()l'CconvClll(OllilI ll(l|it|iIlllIiI~sis WI'iIIO11 I11 lh¢ following wily:
(7]
~llltlftllllh}11 ~,llll','~ll~llt ill
Ill II 1, l~'li'l m l;,I (7)
h:lq h~']vll tti!ltgJil~',r
t W l~md
]?I:
i~ I'~ r~l~tb! 'i,~H m lv
('it I ~lllll III lh~ L'IIIIIIII]L>FII°,,[l;]t~IIl lh, ~Ittti11IlfiliIIll41[ thV } t~Iiil VLt~ ]lii~i'
i'q!qll (qIIIl ('iq~q~If!ILll'ibI
II.eliitllit
J~
ilili lht' '~'IIIIIIUII~ k~:illttIIlI!I IIIu~' i~'~ ~'X;Im
~ :
l~lV filh!
1ti
ltl#i,~i, ft:fl !t,l!Vtt~ i~R
Ill,
I,t,.ti/Lfl
li~
I d~ill
ftlis I,,ilt!,tiflill ~ttltI iilt II ~1 iI ) ~ltlilt it li r 7 f~ Uilil~mi; ~ifiit~ f~!t!l ~;ttI +~ tSJftiliilil tltltl tll~l! tl i ill /JlIH
lll~il
I!,1
tltttt IDt ~'mWV iI!F 1:~ I
I
i,
! ....
I
,
1,~
!
,
I,
I. . . . .
lt(~llilf: ~l{tl ' fi~l:il~ !m i I~iiltl~:l!ii41 !l;ltil i~ ih~: ~:uuiiitl!:~t h!lllt ~,'th t,~ , , ,, , itti!', !tl1 II!~ ill1~;ffc:litill~ hi 111~; m~t!lll~tlt:~]~ t1,I ~t!l!l IIF
I'
!i,
.....
'"l
..... i>,,,
'f ............
. .i...I;i
c:l~:llt/mn;
,
dtl
Ih~ ~ rilk'ill p!linl i~ Ih~li !~.q. (Hi hlih~ Wh¢li ttil 7 7i:(tlil!ll((~ Ill ~!ti~,:Vi(,~ Ih~ 51111t1~ i~L'ill'i~i!Vn p:l~ili(in ~1~ ll) l t(,r I1] It i~ ,lth~l lilllScl f(~r 111 ~r IL } TIle ~>qi¢l s'~t!llCliC'~ i1$¢11 i~ Cllll:lllL:ll~ly il'rL~il~V~lllll Iht Ii, i 171~1~~ h~ "icwl~d a~ llflrillildfdrs I(1 lilt] uqiul;iliil, ~llhil~cl ilnly !o ..~li~fYin!i I ~o ~ltll~r(ll~i~ilc rccliri~ilL'U~. Wiih ihi,~ iil illill(l, <~illC~ I111 illlii ULI <~iilisfv lh~ Sllllli~ I'l~l:llri(;llt'¢ illll.I fl i ril,, 1~< ill~ly hi~ ~lih.slillll{~ll 7(lr i~ ill !h~ fflrluli/ uxi;ic,~siilll (~) iln{l vic~ ;,t:l<~il:
......... , , ' " + " ' 1 ' , ,
~(', ,, ,, > ' " l ~ , , , , , . , . ~ " , f ~
(Ill lht~ l}lhor li~ind, c(lil~iil~l ' h(iw cxchlllll~cl hi cvl]ry lt~l'lll:
~'..
,".......... I"'
" l ' , ..........
(14) i~ c'll~i,l~tl
wlll~ll
', , ~ ~iild ~
~ic ~iilll)l ~,
,
t,', . . . . . . . . , " ' f ~ , , , , , , ~ , , . ~ " , f ,
......... , , , , ' "
~ ",1:',
,~ ,.
the' co~lllcier
'
72
D,L. RusseU
A s a conscqu,.:nce of I , e m m a 3 we can often r e d u c e the numb~'r of e q u a t i o n s in t h e l i n e a r ss!~tems w h o s e s o l u t i o n s d e t e r m i n e the coefficients '~f the c a n o n i c a l s u m s . W e say that (i; . . . . .
i~ ~ , i i, ii~l . . . . .
i~ n, ik, i~,~ . . . .
provicl,ed tilat {j~]- n o d ~,/~} satisfy the s a m e r e c n r r e n u , relalii~n ~lnd tha~ a I = Us, T h e r(rlaii(~n -- i~:
II)(l(I] ~ (ll(l(i), J Illll 1 (IllI~) ~ (ill,i) ~ qllii!), {Ill) I (lltt~+(l(llit
fill]
I)lll,
(ill)
I ' h e i~tlllt.etl datl~l~lik'lil ~L~!I1 i~
~',..,llll'Ill
~ ~'l,,,,I lll[li + u, ll,,j I Ill] I n,,,,j !l I I,
IW~< tt!'ilO iv I hl~toiitl ill' I' I '~¥htql llSill~ redllee(| l!illlilllit'lll terror ) I ql! !tit' ~tlnnlltltltlll ill lilt' .1-11~rlll tile l?'qlfltllllllll i,f J.t'nlnlll 2 t'llll fie t!xl'li'l..ssetl hi I?1ii1~ ill > it'd!it i,d L'ltllinlit!tl[ lel'illS lit hillowt; w,,,,,,lOlllll ~ tv,,.,I Ill1114 u'~.,[I Illl q W~d
II] b[I Ill -
-w,,~.l I I I I ~ w.,<,l ~ ill i u't,d2211 ~ ~,,,H,r2221 !he hliililllil~i [111 I'halll~Jll~ Ihe I,h.a, I~ t.'linllllll:tl' fltrrll mum illlil hi! Wllilen hi It, tlllU'i:tl t'illlllllit'iiJ liir!ll',
I ! i l t - Jlll!l~llllll, I.'--~il I;~STI
3111,i~ ~-Illill÷llll~lt. I I I II ~1 1 iUl ~ I lll~ll ~ltltllll
II,c it'~illlhi~ iil~llli~ Idtlll~ilillil t~l Iho I'li!ilitt'lll~i
II
I
I
IJ
IVlt"j-
l
; hleorrech e, ~e~e~lz,erwes
73
The solution is w ~ , n . - w , l l = ! w~,,,, ~ ') .~;m=~. which i~, consistent with Eq. ('~). In reduced canonic~d term the! ~t~lt :on is written
2 )~ (ooul-
[! Iol -[ono]
[l~l
The system of }il]ear equatinns has been reduced from a .iystem of eight equations to a system of only four eqaaii,3ns. (~s irlore [actors are included ill the ~ttmlnations (m increases) the reduce'on if: ':izc is mnre dlmnatic. The shorlhand notation for the ~t]mmatkm ~lf H~e 4-term )f (,,,~,, ,il.,,k ..... it; as follows (in reduced eanoo'~:td terrns): ~;~.[e~)Ulfi= w,,,r,,.fOnlOl~ w.,.,,[ I(ll)0I I wH,,,,I 110()] :w,,mllIH)l+wm
[lllll,
The: eoildifilln ll~ t~e satisfied I] t2tllllla 2) ~i IhL' !'(i,l~)Wilig;
t~',,,..,,[OlltlOT t wm,~,l If)fll)l t ~ u,, [I l()lt} I w~qt,,[I 110l~ v.h~,[ I ti ij, 111111
w,,~,~l I I I I I ~ tv =,,~,121 I I I t" w~ .,.,~122 e I 1 w ~ , . i 2 2 2 1 1 ~ w,t H122221, The fllilowhl~J fol'intilt!~ Itl'e u~et h~ c!1t111~ctht' I',h,N, t(~ i'e~,hlt'etl f'Hlllllliq!bl Itnrll; 121111
,l[1111t~ I l l Ui],
1 2 2 1 1 1 - ~ 1 1 1 1 t l ~ 311. In I i 1101)], J22211-,I[1111]~311!10]
2[linOl~ImI)t)],
T~le I'Og~.llllll~IlllelU ~3"glL'lll t'4 ~hi)Wll I~eh)w:
lht~rol:~
l)
I
|1
- I
wm, lu
l)
II
{I
2
I I a/ll""
U
I
3
2
I
Wi~m
II ~
[I
(t~)
I1
lift ,~tfltliltlf~l ~11t11} t~(til ~l.~l~'i ~filf [ll~ ~Jtttl tA¢ I~IP 4:l~ffll
~ (i ~11;i)}: (:I'H n~E~
lit.. tC.,ss(ql
74 solve the flfllowing equation:
~. t.,,.,A 0cttl01 ~ .,,,.,,110(10] ~ .,, .,.[ 111)()] - Ut~n.[lll0i+n~i[II/tl] = w,,,..[0tMl)] + w.,r.ll()0(}] t wH,,,[11(/ 4 w,l~,,[ I I ! l i t + wnnll[I I I 1 ]. W e will Ionk for n - n J '¢' such that the above equatk:,n holds. By act c~bvious extension to Lemuna 2 ,~rly .~oh|tioa must satisfy the following condition: w,,¢,.,[(1000' ,~ ~ ~:..[ | (100] + w: t..[ l 1 Off] + w~ ~~,)1 ; ] !0] + w ~ ~[ i I [ 11 ~ n,..,,,[I 111 + n.,.,,[2111 ] + n, ,,.~[22 ! | ] ; ~h, .~,F2221 ] + n H ~, [2222] = - w . . . o l l I I I ] + w.,...[21111 '~ w ~ . , . [ 2 2 1 1 ]
(1 I}
+ w~ ~.,{22211 + writ ~[2222~. It is mos~ c o r w e n i e m to change this equat i on into canon~ca? form with respect to (4 l t I) (instead of w,r,t. (00(i0) as earl i er in this secti, m): [0fl00]-[2227,]-[222l]+[2211]-[2111]+[1.11], [1000]-[2221]-212211]+312111]-411111], (12)
[110i)] = [ 2 2 1 1 ] - 312111]~- 6 [ 1 1 1 1 ] . [I 1 lift
[21113-411111].
T h e rc~ultin~ ~latrix eqt2ation is ~hown below:
2
(}
I
0
- I
(I
()
0
0
I)
wHo~ = -
0
w~t~,,
njw.
.
(13)
trio
-iJLwH,,j
L,Z,,,, J
The illatr'ix is ol rank ./. and we conclude ~flat solutio.., exist, provi ded that Illllll)l ) + ]~ 11 [ {](]ll
tl i VI¢~
2nlll(,+//11
i i = ()
(e~sily derived a:Cler finding the row-echelon form of the matrix [4, pp, 4 8 - 6 2 ] ) , ,'Some of Ifle pos!;ible SUlllroatiolls are the follov.'ing:
31~' {I lontli4
[ I I tilt}-- 211100J4 311 110].
3~!~ {ll0,~lll] 2JOllflll]} -]JltlIlJ, '~ I[lOcllfl- II Ill01 *ill IOl I l I I l l}--[I)O00].
(14)
()ll ihl} eel't hmld side of these equatiollS the stlnllilallds ¢(mt~iSt of I'edtlced c/ttl
nluJlipl¢ canonical terms |llld¢I the ~,,n,.l~:atitn ot~[ not reduced t~lllllrlic[l] (trills) the r.h,s, Ilecd not 12OltSi~[i?[ I'clhlced qllnonil ,I [el'IllS, A silnp[c eMllll[llc i'~ Ihq~ folh'Jw hlg sulnl'Aation:
',~ t( t l(I I l f (O{[ul) ~ (t)l ~ I ) ~ (() 1()0)} : -(()llO)+(()illl+(luO]~
~(1101)
:15J
This summation ~and many others) may be derived by fifth,wing the steps '~f l~q~ (1(1), (111, (I 2) and (135 for tile full system of lb cammical I, rms. The re'm~finl, | 6 x 16 matrix equation (tim analog of Eq. f13), is of rt 'lk t(); E q (15) is just ,~llt: of the p.'/ssible sum nail(ms, Note the close relatio~ ~)1'Eq. (15) and tile first of Eq. ( 14): if all permutaticms of the canonical term!; :if Eq. (15) arc added tot!ether, tilt: first equation of Eq. (14} is obtained. Remember th::; each canonical term in these sums represents the product ¢)f terms from fore distinct sequences (aii satisl}ing the generalized Fiboni~cei recurrence relation' If only one sequence is involved, then some ,ff the equation!~ may be simplified somewhat. Equation (15) becomes the followinR, for instance (in conventi, mal notation):
E (f?d,,,, - f,,f2,,5: :,j:,,, f,. . . . It rema?ns true, however, that tit) cahonicaI s u m exists for ~ f:,
4. Generalized ~ibonacel sequences ' "ontinued) In this section we relax the constraint that a - ( 1 . . . . 1) and allow a to be an m-tuple of arbitrary integer constants a - ( a I, a 2. . . . . a,.5. As in the previeus sectiuu, all sequences {f,} are generalized Fibonaccl scqueno,s defined by f,,, = (,,,, ~+(,.,, 2 £or all n and b - ( b ~ . . . . . b,.,) is an nl-tup~e ol arbitrary integ¢l constants. Consider the slim of the I-term f2,,,,,. We assume thai
Y h,,,,, : ,,,f ...... + ~:f . . . . . . . . and derive the ¢c,ll,~wing condition in horthand notation (Lemma 2): c~;(O) + e~(I)
F (2) - (',,(25 + e,(:!L
Using the relations ( 3 ) = (0)+ 2(1) and (25= (()5+ (1) wc find tile r(~llowbll~ systcnl of cquation~:
It, [
[:'ll:
tile solution is ~,, fl. c I ]'~ '['l
Ill,
I. '['herefm'c we have the frith*wing sllltllttnli(n.:
%
tll
l'he s~me prn~pdm~ Iolh*wh~g forlntlh~h:
I~lmo¢l
~pplletl In lhe .c~ps wh, '
-(3), ~4),.., leaO~ to lhe
2 ~ t~,,,,, ~ f ...... ~ f ....... I = f .........
(16)
and ill general
(Lk
1 ( i)"}~'.f~,,,,,-(bi,
, - ( - 1 ) " ) f k . + , . + F ~ f , ...... ,
= (~,(,,, ,,,,, - ( - l)"fa ...... where { L j is the sequence of Locas nmnbers (L~.- L~ t~ Lk z, /-.o-2, L~ = 1) ard {F~,} is the sequence of kibonacci numbers (Fk =F~ t+F~ ~. F ~ = 0 , FI = 1). The ]asl ..quation of ( t 6 ) u~,;es the well-known identity [~,+~-.F'~ ]f,,~+F~[,.,, which is easily verified. Com;idcr the sum r.f,,.,g_.,,..; the canonical terms in shorthand notation are ([tl]h (01), (10). and (11), The condition to be satisfied is the following:
c(,.(O0)+cm(Oi)+em(lO)+eu(ll)+(12)
=
- ca,(12)+ cm(]3) + cm(22)+ cu(23). After changing tile r.h.s, to canonical terms and solving the associated lineal S}'tcln. the tanonica] SUm is found: 2/.V !t)0) = ( I ) l ) - ( 1 0 ) + (1 l), (171
or
2 2 f,,~,g ..... = f . , . g . . . . . . . - f . ~ , g z , , ~
+ f.-,,tg .......
tThis ~am may be simplified somewhat.) TI',e same teehni~ .'s lead to tl-,e conclusions that the sums ~ fz.+.g2.+, and ~ L,~,, ~.h~,,,, caanot be expressed as canonical sums. In both cases the matriee... of the assodated system of linear equations are singular matrices and no solutions exist t,.) the systeras.
5. General recun'ence sequences
The methods of this paper can be applied to a wide variety of situations. Consider the Kzneral second order recurrence sequences {r.} and {s.} defined by
r,,=ar,, t+br. 2 and
s,~=es.-~+ds. 2.
The cc, nd~lion of Lemma 2 can be reduced to canonical form with the following
((2)=~(111~ #(!(1),
(21t--~(ll)+h((ll), (22) = f,c(l li F ~hl(]',~)+
bc(Ol)+fid((lO).
T~le restllth~g linear eqtlation is a~; i'oiinws:
11 I
1 -d ,-
b 1 -n
h( b',,,t = [il)]. / r -mll|ll~,,t !,-ac~j,:,,J
(i8)
,
If a, h. c, and d are given, the solution to (18) defines a canonical sum for r~,+~s.,.. For enan~ple, if
r,,=4r. ~+r,, ~
s.=-Ss,, tks,, 2
and
the resulting summati(.~n is ~ r.s. - - r . ~is~ - r,,s~,+ i,
If a - O , b
=
~r,,s,,
! and
c: ( d -
1)2;~ 0 the retulting s u m m a t i o n is
1
c2 (d_l)z[d(l --d)r.s.+cr.s..~+cdr.÷~s.+(1
d)r,,,~s,,.~]. [
Ug)
Initial conditions of ro = t. r ~ - - 3 give the s u m ~ . . . . . . ~ ( - l ) " s . ; akernatively, initial conditions of ro = 1, r~ = 0 give the sum ~ z i ~ s~i. T h e same m e t h o d s also apply to h i g h e r - o r d e r recurrences. As a final e x a m p P , let {t.} be the sequence of generalized Tribonacci n u m b e r s defined by t. = ~._~ ~-t. : 4 ~-3. It is easy to derive the following ~um: 2 ~ (0) = ( 0 ) + ( 2 ) or 2 ~ t,,+p = t.+. + in+p+ 2.
6. Criteria for non existence o| the s~a~lard stun Several of the s u m m a t i o n s conside;ed in this p a p e r could not fie ex ,rrssed as s t a n d a r d sums. T h e reason for the non-existence of standard sums ca : Oe better u n d e r s t o o d if the solutions to the individual recurrences are wrilten u terms uf the roots of their characteristic polynomials. Let a~ oe any root of the c Faractedstic polynomial of the sequence {f,} and ~ssume that a = ( l , 1 . . . . . t). T h e n suppose for Ihc m o m e n t that all of the roots are ~,i~tinc| and consider tire expansion of tile m - t e r m B(n,a,b,j). O n e of the terms in the s u m m a t i o n is
lit~ illwil%, lllie~ hld~lilflI~ hqlilltliflJiqlIttH,Sl lh~ r~'.,si ~,~ IIfl lh~ lJlllJl~ fl - ~/ ~tltll
II
~.
II .41111pll~: Ihld ~-'ltellJFJ~llts Vl ~ J M ~tl~qt ,
IIIL~ llllhlwJllB CllltdJlilllt is
l']leI: iiilch ti~rIll llf i]!~J Stlllllllilliilllo! the nl-teFnl Cltll lle iedut't:d Ill it Iiae~ll' Clnlqqilation of prildu!:Is ~I' rilOls, hl tinll Jh~} SHill ill these pro'JtlClS n[ roots can h~2 [ilChll'ed inhl p r o t h i . l s ill tile 1"12CnlTcnk'~gt e r m s ; the resulI is ~ite l'antnliciil SHill. It mnlliple r n m s esisI o r if a .~ {1, I . . . . ,I, Ihe p r o c e d u r e is s l i g h t 1 ) d i f f e r e n t , hut tile e n d resuh is the Sltllle, T h e m e t h o d o l o g y described hi Sel tkm 2 and illustrated ill Suclions ;! 5 is essclltially a wily to d e t e r m i n e tile cneilicien~.s q wJihollI havin~ Ill lhld Ihe r~lllts of tile cnilrilClelisiJc eqtlati(ms, [l ~ ¢ t e . . e~,,, - 1 Ior s o m e set of foists ~ , , . . . , ~,,,, the slalldard s a m doe!i I1~.~1 exist, hi lh[ll cilse the.: ~llqerm tO be sal?lnlCd contain~i a t l l m Of t h e form ' m,,]' = ~ c( I t = on,
V
which ca!t~lol he I'acR~red into a p r o d u c t i~f r e c u r r e n c e ternls. This is illustrated in m o r e deill]I fol s e c o n d - o r d e r r e c u r r e n c e s i i {51. SittialJolls encounh.'red hi this p a p e r w h e r e no s t a n d a r d s u m exiats can now he.' e×phfined ~ e t If,,}, {g,,}, {h,,~,, {k,,} be F:bonacci s e q u e n c e s ; t h e c h a r a c t e r i s t i c eqtl~tlI~lll is ~2 -X-- 1 = 0 and the roots are
~:,~ll
~/?=
1.61g,.,
and
,~=~(1-.].~)=-0,618
... .
T h e sun:.; ~_~[,,g.h.k,, ~ (.,,g.~,,, and ~ , , g , , h 2 . all o w e t h e i r n o n - e x i s t e n c e es stand~lrd sums to the fact that & : ~ 2 _ 1. In t h e case cff E q . (19), t h e c h a r a c t e r i s t i c polylvmfial of {r,,} is v 2 - 1 = 0 w h o s e r o o t s are 1 and - 1. T h e s u m m a t i o n does not exist as a s t a n d a r d sbm if t h e c h a r a c t e r i s t i c p o l y n o m i a l x e - ex - d = O ok Is,,} has a root that is e i t h e r I e r - 1. If c -- d - I = 0 t h e r o o t s are c - 1 a n d 1 ; if ? - d + 1 = 0 l h e roots are c ~ 1 und - 1 . T h e s e a r e precisely the cases in w h i c h (19) does not hold N o t e that it i! crit(cal that a s t a n d a r d s u m be valid for all s e q u e n c e s satisfying tl~e n-ecurrence e l a l i o n . Oeherwise t h e initia| c o n d i t i o n s could i,e such t h a t t h e t r o u b l e s o m e t e r m de.es not a p p e a r in the nl-'~erm b e i n g s u m m e d . For e x a m p l e , let {.-I, f,-d,, f,,-L ,+[. ,.Then 4,, ~4 f.:&" and ~ f , ~ : ~ & a " = ~ = c f a + ,.
T}I0 hld~lJllJl~ ~HilIltl!llh)fl I~ ~lt~)ihlr lu lh¢ ql mt~id, ~lll IhJq J~ HoI (ill iflM)lil(-:o ul l) t'ttltltllJtJ)ll ~llltl IleQfl')~,~ jl j~ lift( v01hl fill: qli o IIClW~I htt!ihl:Vilt~, ti~ -- 1], I ~ ~:
(IIWHI It ei~l IH" ~OIJltdl!L'~ d~JILIrlliHed hy lih,~it[le; IlllelIC~ HJh!i(III~ [hi~ I):LII;~I" V~lllhll'~:s4 nlelhc~di~hlgy for ~hlding III¢ indelillih.i(fllldd~qhfilei htllIl~eli'pr,),lucl~, (i[ telIllSIfIllltlJl~ ~e'qtli:ll¢~ YlP~ ~tllt~s~(l [¢iiiiIiiil[e dihlillclJvc illseveI~II Wil~,: lit IIlny eo6~jsl o[ it lJq~aF L'onlhJilqlJiln ~[' IeIIllh, called c~IIl(IIlJt'iIIleIlI]~i (•) eiICJlcIIIlllllJ¢[l[t,atm ctlll~i~it~(i[ d pO)dllCl ol fib Itllh,iml. l~it+h)l[r(1111~+'+i:ll,ff lhI.~ origin~d ~;equellCeS; (3) Ihe ~uhscrJpIs ()l" Ihe filCtOl'S at,' lillUar tuiwIhlllS eli Ih¢ ~Hinnla~hm Wlliahle, (4} tile [dentifie~. (ff Ihe original heqnences are nlainhlinucd in the final lind:hi ~'iml[lJllal Joll ; alld (5) the SLIIllS ill'(3 valid for all SCqLh~'IlCeS hatisfyinB the FectlFCeIlCC ,-ela!J(llls. The rnetltod finds Stlch a IJllear co:t]tlJllatk)n wfienever one i.xJsts and ;llay I,e SgllllllarJzed LI!I fol) )WS: ,vrite the e q u a t i o n of the indue!ion 'step ([.¢Fiuri~l 2): (2i find a se) of canonical ~erm~ and write the c,~,,mions of ;tep ( I ) Jq terms of lfies ! canonical telms (ill some ease~ red:+ced canor;ical ~cl-i-ii~ may ft.' used): (3) write the ass ) c i a t e d system of linear e q u a t i o n s (this ~yslem must exist since t h e canonic~d t e r m s a r e linearly i n d e p e n d e n t ) ; (4) solve the linear e q u a t i o n s - - t h e system!: of linear e q u a t i . m s has a '~olution if a n d only if l h e d e s i r e d indefinite sun, exists as a linear c o m b i n a t i o n of canonic.M t e r m s ; and (5) simplify as desk'ed. (i;
8. References [1] G Berzsenyi, Sums cff wnducts of generali:'ed Fihonacci numbers. Fiblmacci Oua:l 13 (1975) 343-344 [21 J,C. Pond, Generalized Fihonacci surnmafi,~ns, Fibonacd Quart, 6 ([9681 97 II)8 [3] Dr_ Russell, Notes on sums of produch of genelali/ed Fihoqacci numbers. Fibor ,cci Quart. to appea~, [4] E,D Nering, Linear Algebra and Matrix 1 ory (J, Wiley, New YtJrk. NY, IC)a3L [:~] D,L. Russell, Summation of second-order ,,'currence terms and their squares. Fiboneco Ouart, Io appear