ON THE PARTITION OF HORADAM'S GENERALIZED SEQUENCES INTO GENERALIZED FIBONACCI AND GENERALIZED LUCAS SEQUENCES

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1 ON THE PARTITION OF HORADAM'S GENERALIZED SEQUENCES INTO GENERALIZED FIBONACCI AND GENERALIZED LUCAS SEQUENCES A. J. W.HILTON The Uiversity of Readig, Readig, Eglad 1. INTRODUCTION If p,q are itegers, p +4q 0 f let co = co(p,q) be the set of those secod-order iteger sequeces (wj = (w 0, w v w 2, ; satisfyig the relatioship which are ot also first-order sequeces; i.e., they do ot satisfy W = cw 7 fy ) for some c. I Horadam's papers ([3], [4], [5], [6]) our W is deoted by W (a,b;p,-q). I this paper we show that oo ca be partitioed aturally ito a set F of geeralized Fiboacci sequeces ad a set L of geeralized Lucas sequeces; to each F e F there correspods oe L e L ad vice-versa. We also idicate how very may of the well-kow idetities may be geeralized i a simple way. 2. THE PARTITION OF oo(p,q) If a,(l are the roots of x -px -q = 0, d= +\/p 2 + 4q the the followig relatioships are true: a*fi = p, aft = -q, a - j3 = d, (1) w^^f.. where A = W 1 -W 0 & B = W 1 ~W 0 a. Sice (W ) is ot a first-order sequece it follows that a±0,$±q. A * 0, B J=0. Whe W is represeted as i (1) we say that W is i Fiboacci form. O the other had, with differet costats C ad D, W could be represeted as W = Ca" + D$. I this case, we say that W is i Lucas form. Whe W is i Fiboacci form (1) we may perform a operatio ( ') to obtai a umber W' m where W = Ad + B$. We say that the sequece (W' ) is derived from the sequece (W ). The sequece (W' ) is a sequece of itegers sice (2) W Q = A+ B = W 7 -W 0 $+W 7 -W 0 a = 2W 1 -W 0 (a + $) = 2W r ~pw 0 ad (3) W 7 - Aa + B$ = (W 7 - W 0 $)a M\N 1 - W 0 a)$ = W 7 (a + &)- 2W 0 a$ = pw 7 + 2qW 0. W' may ow be expressed i Fiboacci form. I that case If we perform the operatio ( ') o W we obtai W% = [Ma - p)j a + f(-b)(a - fi)j $ -'«-«2 [^f] ifw 339

2 340 OSy THE PARTITION OF HORADAMTS GENERALIZED SEQUENCES INTO [DEC. We have proved Theorem 1. W% = W for ail > 0. St is ot hard to verify that the equatio W = W' (v ) caot be true if (W ) is ot a first-order sequece. Throughout this paper let (X ), (Y ) e co (p,q), let X' =Y ( = Q, 1, 2, - ) ad let X 0 = a, X 1 = b. The, from (2) ad (3), YQ = 2b - ap, Ky = pb + 2qa. By theorem 1, therefore, or directly, it follows that a = 2Y 1 -py 0f b = py, +2qY 0. The followig theorem ow follows easily: Theorem 2. (i) (4) \2Y -py m.t ad \py +2qY _ 1 for all >1. Proof of (ii). If (ii) If (W )eo>(p,q), \2W 7 -pw 0 ad \pw 1 +2qW 0 the (W ) = (X ) for some (X )e<a)(p,q). X 0 -^Lzfo. Xl.»bimL a d (x H,e»< M ). the* ', «2 xbjp^^oy p hw^\ Wo ad x,j P w 1+ m 0 \ +2qhw^wg\ =Wi which proves part (ii). The basic liear relatioships coectig (X ) ad (Y ) are described i the followig theorem. Theorem 3. The followig are equivalet: ( (X' ) = (Y ), (ii) Y = 2X +1 - px for ail > 0, (iii) Y H = px +1 +2qX forail >0, (iv) Y = X +<i + qx _ 7 for ail > 1, (v) X = 2Y +1-py f f l r a >Qf (vi) x +1 = py+1+2 SXR foras >Qf (vii) X = Y -H + l +qy»-l for all >1. NOTE: For each of (ii),, (vii) we eed oly require that the expressio is true for two adjacet values of. Proof (i)=*(ii). If (X' ) = (Y ), the from (2) ad (3), Y 0 = 2X 1 ~px o ad K ; =px 7 +q2x 0 = 2X 2 -px-j sice X 2 =pxf + qxg. Let m>2 ad assume (ii) is true for 0<<m. The Y m = P Y m-1 + Q Y m-2 = P(2X m -px m^) + q(2x m^ -px m 2 ) = 2X m+1 ~px m. The result ow follows by iductio. (ii) ^(iii) <» <*» (vii). This follows easily usig X +1 = px +qx ^ ad Y +1 = py + qy ^ (>f). f(iiuiii),-(vii)]=>(i). Sice x 0 = ^fxo ad Xf_Ph+*yo it follows from (2) ad (3) that

3 1074] GENERALIZED FIBONACCI AMD GENERALIZED LUCAS SEQUENCES 341 ad similarly X^ = Y-j. Hece (X' ) = (Y ). This completes the proof of Theorem 3 We ow describe the partitio of oo(p,q) previously referred to: If (W )<Eto(p,q) ad d * 1 let W = d zm to for all > 0, where r^o is a iteger, (co ) G CO ad d / oo for at least oe > ft The If (W )<=oj(p,q) ad d=1 Set (W )<E L if \2cjf -pojg ad \poj^ + 2qu>Q, (W )^F if either ^2co 7 -poo 0 or ^pojf+2qco 0. (W )^L if WT-WQGLKO, (W ) ^F if W 7 -W 0 a > 0. The assigmet of (W ) to L or F is atural i the case d/1, but if d= 1, although the partitio itself is atural, it is ot true to say that a sequece is "like" the Lucas sequece rather tha the Fiboacci sequece or viceversa. I view of Theorem 3 if (W ) is a member of F {or D the ay "tail" of (W ) is also a member of L, respectively). Theorem 4. (XJ e F if ad oly if (Y ) G L. Proof Casel. d=1. (X )e F o X = Aa - BP, where B < 0 o Y = Aa ~ (YJ G L. + B$ Case 2. d?1. (i) If (X ) G.F suppose that X = m x for all > 0, where m>0 is a iteger, fr J G F a d tf 2^ for at least oe >0. Clearly </ 2 K*0 or flf 2^;. By Theorem 3, Y 0 S= 2X 1 ~px 0 ad Y 1 =px 1 + 2qX 0. Let Y = m y forail /? > o. The y# = 2x 1 -px Q ad j / v =px t +2qx Q. Sice 6r J G i^ either \2x 1 -px 0 or \px 1 +2qxQ. either / 2 Vo r ^ J W - But it is easy to verify that 2y 1 -py 0 = x 0 ad py 1 +2qy 0 = x 1. ( / j G L ad so f K^J G L (ii) If (YJ<EL suppose that Y = m y forail /7 > 0, where /?? > 0 is a iteger, /> ) G L ad / 2 / for at least oe /? > & Clearly ^ 2 K ^ or \yp By Theorem 3, F(m Let * = m x 2Y 1 - P Y 0 X 0 = for all /7 > 0, The 2yi~PVo * 0 = ' ' _py 1 +2qY 0 A / - X1 _ PYi+2qyo Sice (y ) ^L, so * 0 ad * / are itegers, so \2y 1 -py 0 ad \py 1 +2qy 0, (x )^co-.but 2x 1 ~pxo = yo ad pxj+2qxo = Yi, ad sice \y 0 or \y 1 it follows that either )(2x 1 -px 0 or \px 1 +2qx 0. (x )(=F ad so (X )(=F. This completes the proof of Theorem 4. Here are some examples of members of F alogside the correspodig member of L. 0, 1, 1, 2, 3, 5, 8, 13,.» 2, 1, 3, 4, 7, 13, 0:i,p,p 2 + q,<- 2,p,p 2 + 2q,- 0, 1, 3, 7, 15,, 2-1, 2, 3, 5, 9, 17, -, 2 + 1, - 0, 1, 2, 5, 12, 29,. 2,2, 6, 14, - (Pell's sequeces) a, b, qa, qb, q 2 a, q 2 b, 2b, 2qa, 2qb, 2q 2 a, 2q 2 b,

4 342 ON THE PARTITION OIF HORADAM'S GENERALIZED SEQUENCES INTO [DEC. 3. BINOMIAL IDENTITIES May idetities ivolvig Fiboacci ad Lucas umbers are readily derived from the biomial theorem; for example see [1], [2] or [8]. They ca early always be geeralized to become idetities ivolvig geeralized Fiboacci ad Lucas umbers, I this sectio we could derive a log list of such idetities; but this seems uecessary i view of the proofs i [21 ad [8], ad also it would take up a lot of space, as the costat multipliers which have to be itroduced seem to make the geeralized formulae up to twice as Sog as the formulae i [2] ad [8]. Istead we derive oe set of idetities as a example ad show how further idetities may be derived. There ofte seem to be two very similar idetities, oe featurig Fiboacci umbers, the other Lucas umbers. Whe there are two such idetities they may ofte be derived from oe idetity by usig the fact that 1 ad y/e are liearly idepedet over the ratioals, although this is ot the procedure adopted i [21 or [81. With geeralized Fiboacci ad Lucas umbers such a process would ot be appropriate, but, as the examples show, the method of proof which is atural does lead to a sigle idetity, from which the two idetities may be obtaied by specializatio. For this sectio (F ) ad (L ) deote a pair of sequeces such that (F )^F, fz^jez, ad (F )' = (L ). Also, C=F 1 ~F 0 P, D = F 1 -F 0 a. The atural method of proof is firstly to derive a sigle idetity ivolvig (X ) ad (Y l The either of the followig sets of substitutios may be made:! X ~ F+r (The third of these follows sice ad the fourth follows similarly.! Or Y = L + r A = Xj- X 0 P = F rh - F r p = Ca r B = Xj - X 0 a = F r+1 - F r a = Dp r. "~r+i ~ Ca*1 ~DP r+1 = (Ca r -DP r )P~-Ca r p+ca r+1 = F» +Ca r a-p a-p X - L +r Y = F +r A = Xj - X 0 P - Lr+j - L r P = Cda r B = Xi-X a a= L r+1 - L r a - -Dcip r. The each of these sets of substitutios leads to oe of the two derived idetities metioed above. EXAMPLES OF BINOMIAL IDENTITIES EXAMPLE I Sice a m Y, + dx m =.JH 2A 2B it follows that, ' ri'y 1 Y " s a A m r m (?)- ad p m = mr J^ f - / j V ' 4 C ( i- t i=q Y m + dx m = (2A) U ] T d'x E "'*!» l m YST\ Y m-'> (?), ad Y!?) m - dx m = (2B) U l-lfd, X l my^1 (,")., X m = 2-"d~ 1 {dxjytfitf) [A 1 - -(-1) i B 1 - f> ].

5 1974] GENERALIZED FIBONACCI AND GENERALIZED LUCAS SEQUENCES 343 A similar formula may be derived for Y m. Makig the first set of substitutios, we obtai But F m+r = 2-»d" 1 Y. (df m+ rh ^r (?)([Ca r ] 1 ' - (-1)! [Dp] 1=0 pl-^r-r / 1u'1-r-r _ r1- \ * r-r + *r-r \ \ VD (L r r dr C a -1-7)0 p -C r_ r Yc I 1 ' ) 2 \ cr Dr J z \ c Q \ F m+r - r - ' f 1 t ( d F m + r H ^ 1 ) J L r. r [ ^ - ( - D ' j - ) + df r-r{j- -(-^j- )\ Makig the secod set of substitutios we obtai L m+r = 2~"d- 1 J^ (dl m+r ) i ( F m+r r i ( 1 ) (fedatj h ' - (-in-ddpl 1 '") = 2 ~ E ^ W c t ( ; j (rearj? ~" - (-»"-' (- 1 - wi 1 ~ ) But = 2 ~ (df m+r ^+r ( i ) <ica r ] 1 - +(- irw 1 ' ). C1- ar-r + (_ 1)iD1- rm ^ B ( J L + {-tf JL j +dfr. m ( ± -(-,)< J- ). \ J EXAMPLE 2. Sice dx m = 2Aa m - Y m ad dx m = ~(2B$ m - YJ it follows that akd " X m = E (-Yj(2A) - i [ j\a m ' mi+k ad fd X% - (~1) li/odi-i ( \ (-YjW) [^ff 1 om-mi+k ad Y k d X" m +X k d +1 X" m - 2. <-Yj(2Ar i ( [. )(Y m mj + k + dx m mi + k ) s=0 Y k d X m-x k d +1 X" m = (-1)" (~Y m ) l '(2p) '^ 1=0 f )ly m mi + k -dx m mi+k ).

6 344 ON THE PARTITION OF HORADAM'S GENERALIZED SEQUENCES INTO [DEC. X k X m = (-i^"- 1 [ \ [Yr-wHkfA"^-(-1) r') + dx m. mi+k (A - i + (-1)"^-')] ad Makig the first set of substitutios we obtai ad zt/ 7=0 ^ = ^ r Z t-d'l-m 2 "' 1 [l)ll m -mi + k(c -'' + (-1)" D"' 1 )+df m mi+k (C --(-7)"D"-')]. i'=0 Makig the secod set of substitutios we obtai F k L" m = j ^ (-V i ( Fj'2 '(»)f t m^(* 'C^ + {-i-v^d^d' H ) 2d " M> so that + dl m. mi+k (d - i C"- i - (-1) (-1j - i d - i D"- i )] ad L k L m = j^ (-i Fj2 - i [.)[ F m. mi+k (d"- i C - i -(-1) (-1) - i d"- i D - i S so that + dl m. mi+k (d -'C -' + (-V (-l) -'d -'D -')J L k L " = 1 Z ^ m / ^ " ' (? ) ll m - ml+ k(d - i H-1) l C - i ) - df m, mi+k (D -'' - (-D'C"' 1 )]. Further three term idetities from which biomial idetities may be derived i the way described are dx = Aa" - 50", Y = Aa" + B$, (5) Aa m+ = X m a +1 + qx m. 1 a, (6) Bp m+ " = X m?p +1 -t-qx^f, a 2 = pa+q, f ' = pp + q, (7) Y 2 = X 2 +4AB(-q)i, Aa 2m = Y m a m -B(-q) m, Aa 2m = dx rr fll m + B(-q) m Bf m = Y m $ m - A(-q) m, B0 2m = -dx m & m + A(-qj m.

7 1974] GENERALIZED FIBONACCI AMD GENERALIZED LUCAS SEQUENCES 346 Most of these idetities are obvious, or early so. Idetity (5) may be proved as follows: Aa m = V 2 Y m + y 2 dx m = y 2 ( P X m +2qX m 1 +dx m ) = X m (< L j il ) +QXm-l = X m a^qx m^, ad idetity (6) is proved similarly. Idetity (7) is proved as follows: Y 2 = (Aa + Bf) 2 = (AcP-B$ ) + 4AB(a$) = (a-$) 2 I ^ " ' ^ ) + 4AB(-q) = X 2 +4AB(-q). ACKNOWLEDGEMENT I would like to thak Professor A.F. Horadam for poitig out a error i a earlier versio of this paper. REFERENCES 1. L. Carlitz ad H. H. Fers, "Some Fiboacci ad Lucas Idetities/' The Fiboacci Quarterly, Vol. 8, No. 1 (Feb. 1970), pp V.E. Hoggatt, Jr., Joh W. Phillips, ad H.T. Leoard, Jr., "Twety-Four Master Idetities," The Fiboacci Quarterly, Vol. 9, No. 1 (Feb. 1971), pp A.F. Horadam, "Basic Properties of a Certai Geeralized Sequece of Numbers," The Fiboacci Quarterly, Vol. 3, No. 3 (Oct. 1965), pp A.F. Horadam, "Geeratig Fuctios for Powers of a Certai Geeralized Sequece of Numbers," Duke Math. Joural, Vol. 32,1965, pp A.F. Horadam, "Special Properties of the Sequece W (a,h; p,q)," The Fiboacci Quarterly, Vol. 5, No.5 (Dec. 1967), pp A.F. Horadam, "Tschebyscheff ad Other Fuctios Associated with the Sequece j W (a,b; p,q)\/' The Fiboacci Quarterly, Vol. 7, No. 1 (Feb. 1969), pp Muthulakshmi R. Iyer, "Sums Ivolvig Fiboacci Numbers," The Fiboacci Quarterly, Vol. 7, No. 1 (Feb. 1969), pp H.T. Leoard, Jr., "Fiboacci ad Lucas Number Idetities ad Geeratig Fuctios," Sa Jose State College Master's Thesis, Jauary, ERRATA Please make the followig correctios to "Fiboacci Sequeces Modulo /^/'appearig i the February 1974 (Vol. 12, No. 1) issue of The Fiboacci Quarterly, pp O page 52, last lie, last setece, chage "If 2/f(p)," to read "If 2jff(p)/ f O page 53, chage the fourth lie of the third paragraph from "which (a,b,p e ) = 7,"to: "which (a,b,p e )^1 9 " O page 56, third paragraph of proof, teth lie should read: "...is give by 5 2e - 5 2e ~ e " 2 = 4»5 2e ~ 1..." O page 61, chage the secod displayed equatio to read: (k) = 2 p?. k Lie 7 from the bottom should read: " f o r i=t,.», * - 1. "