SOME RESULTS ON FIBONACCI QUATERNIONS MUTHULAKSHMI R. IYER Indian Statistical Institute, Calcutta, India

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1 SOME RESULTS ON FIBONACCI QUATERNIONS MUTHULAKSHMI R. IYER Idia Statistical Istitute, Calcutta, Idia 1. INTRODUCTION Recetly the author derived some results about geeralized Fiboacci Numbers [3J. I the preset paper our object is to derive relatios coectig the Fiboacci Quaterios [ l ] ad Lucas Quaterios 9 to use a similar termiology, with the Fiboacci Numbers [2J ad Lucas Numbers [4] as also the iter-relatios betwee them* I Sectio 3, we give relatios coectig Fiboacci ad Lucas Numbers; i Sectio 4, we derive relatios of Fiboacci Quaterios to Fiboacci ad Lucas Numbers, ad i 5, Lucas Quaterios are coected to Fiboacci ad Lucas Numbers. Lastly, i Sectio 6 are listed the relatios existig betwee Fiboacci ad Lucas Quaterios. 2 TERMINOLOGY AND NOTATIONS Followig the termiology of A F. Horadam _lj, we defie the Fiboacci Quaterio as follows: Q * F + i F + l + ^F k F : +3 where F is the Fiboacci Number ad i,j,k satisfy the relatios of the Quaterio viz: i 2 = j 2 = k 2 = - 1, ij - ji = k; jk = -kj = i; M = -ik = j. as th Now o the same lies we ca defie the Lucas Quaterio T say T = L + il JjL, Q +2 + kl +3 _,_ Q th where L is the Lucas umber. Also, we will deote a quatity of the form 201

2 202 SOME RESULTS ON FIBONACCI QUATERNIONS [Apr, F " i V l ^ F - 2 " k F - 3 by Q * ad F + IF - + Jj F 0 + kf Q by Q Similar otatios hold for T + ad T -, that is s J ^ * L - IL + jl 0 - kl Q = T # - 1 J -2 ~3 * ad L + IL. + JjL. + kl 0 = T ~3 Now we proceed to derive the relatios oe by oe. All these r e s u l t s a r e obtaied from the defiitios of Fiboacci Numbers ad Lucas N u m b e r s, give by, F = l^jl., L = ( a» + b } V5 for all» where a ad b a r e the roots of the equatio x2 _ x _ i = o, obtaied from the Fiboacci ad Lucas r e c u r r e c e relatios. coected by The roots a r e a + b = 1, a - b = y/e 9 ad ab = 1

3 1969] SOME RESULTS ON FIBONACCI QUATERNIONS 203 Cosider the followig relatios: SECTION 3 (1) F L = F +r +r 2+2r (2) Therefore (3) (4) (5) F L = F r - r 2-2r F -L _, + F L = F Q L 0 +r +r - r - r 2 2r F -L _, - F L = F 0 L 0 +r +r - r - r 2r 2 F _, L = F 0 + (-l) ~" r F 0 +r - r 2 2 r (6). F - r L + r F 2 " ( "" 1) F 2 r Therefore (7) F ^ L + F L _,_ = 2 F 0 +r - r - r +r 2 ad (8) (9) (10) F ^ L - F L ^ = 2 ( - l ) " r F 0 +r - r - r +r 2r F _, L = F 0 _ + ( - I F F +r 2+r r F L _, = F Q _, - ( - I F F +r 2+r r So (11) (12) F ^ L + F L ^ = 2 F 0 a. +r +r 2+r F ^ L - F L ^ = 2 ( - l ) u F +r +r r (13) F L = F + (-1) F v ; +r +s 2+r+s r-

4 204 SOME RESULTS ON FIBONACCI QUATERNIONS [Apr. (14) F L _,_ = F,, + (-l) + S + 1 F +s +r 2+r+s r~s (15) F ^ L ^ + F ^ L ^ = 2 F O J L +r +s +s +r 2+r+s (16) F ^ L ^ - F ^ L ^ = 2(-l) + S F +r +s +s +r r - s SECTION 4 I this sectio, we give the list of relatios coectig the Fiboacci Quaterios to Fiboacci ad Lucas Numbers. The simplest oe is (17) Q - i Q j Q k Q + 3 = L + 3 Cosider (18) Q - l Q + l " < = ( - D [ 2 Q i - 3 k ] ( 1 9 ) < - l + < = 2 S - l - 3 L (20) Q ^ + J - Q ^ _ 1 = Q T = ( 2 Q 2-3 L ) + 2 ( - l ) + 1 ( Q 0-3k). ( 2 1 > V2 Q -l + V W " 6 F V l " 9F ^ ^ V l ) " 3k > (22) %-!% * " < + l = ( - «[ 2 + «+ 3j + k] (23) Q - l Q + l " Q -2 Q + 2 = ^. [ " o " k ] + *<rlf +1 [ Q, - 2k] < 2 4 ) Q -3 V 2 + Q Q + l = 4 Q 2-2 " 6 L 2 + l < 25 > Q Q + 1 = 6 F + l V l " 9 F ^\-2) Also the remarkable relatio (26) g + r + ^ - 1 ) Q - r = Q

5 1969] SOME RESULTS ON FIBONACCI QUATERNIONS 205 ( 2 7 ) Q + l - r Q + l + r " Q + 1 = ^ ^ ^ ^ 2 ^ % " 3 r > 1 Now we tur to relatios of the form: (28) Q + r L Q + r - Q r + ( - l ) + r Q 0 ( 2 9 ) Q - r L - r = Q 2-2 r + ^ ^ O ( 3 0 ) WW + Q - r V r = Q 2 L 2 r + ^^"X (31) Q L, - Q L = F 0 T 0 +r +r - r - r 2r 2 (32) Q ^ L = Q 0 + (~l) "" r Q 0 7 Ti+r - r ^ 2 2 r (33) Q L _, = Q 0 + ( - l ) ~ r + 1 Q 0 * 7 i i - r +r ^ 2 ^ 2 r * (34) Q _,_ L + Q L _ L = 2 Q 0 + (-l) ~" r L 0 Q A ' ^ + r - r ^ - r +r ^ 2 2 r * 0 (35) Q ^ L - Q L ^ = ( - l ) " r x F 0 T ' ^ + r - r ^ - r +r 2r A 0 (36) Q, L = Q 0, + (-D Q v v ' ^ + r ^ 2 + r ' ^ r (37) v ; Q L = GL - x (-l) ; Q* ^ +r ^ 2 + r ^ r (38) Q ± L + Q L, = 2Q 0 _,_ + ( - l ) v L Q ' ^ + r +r ^2+r r A 0 (39) Q + r L - Q L + r o ( - l ) - F r T 0 ( 4 0 ) W t = Q 2 + r + t + <-» + \-t < 4 1 > V t L + r " Q 2 + r + t + ( " 1 ) + r + l Q ^ Therefore; (42) Q j L ^ + Q. J L a. = 2Q 0 ^ L, + ( - l ) + t L J 3 A v ^ + r +t TL+t +r ^2+r+t r - r * 0

6 206 SOME RESULTS ON FIBONACCI QUATERNIONS [Apr. ( 4 3 ) Q + r L + t " Q + t L + r = ^ X ^ O (44) Q + r F - r ~ 5 [ T 2 " ( - 1 ) " r T 2 r ] (45) Q - r F + r = 5 [ T 2 " ^ " ^ a r ] ( 4 6 ) V r F - r + V A + r = k [ 2 T 2 " ( - 1 ) " r L 2 r T o ] < 47 > < W F - r " V r F + r = ^ ^ r S ) (48) W = i [ T 2 + r " ^ r ] (49) Q F + r = I [ T 2 + r " ^ l ] (50) V + V m x = \ [ 2 T 2 + r " < - «V o ] (51) Q ± F - Q F ± = (-l) + 1 F Q A ^+r ^ +r r*0 (52) Q +r F +t ~ 5 [ T 2+r+t " ( " 1 ) Q T r - t J (53) Q +t F +r = 5 [ T 2+r+t " ( _ 1 ) Q * T r = l J (54) (55) (56) W-r Q +r F +t + Q +t F +r = 5 [ 2T 2+r+t " ( _ 1 ' L r - t T o ] W * " W + r = ^ X - t S ) + ^ X - A + r = I [ T 2 ( 1 + < "» ' >" ^ ' ^ r " < " «X ] r-t

7 1969] SOME RESULTS ON FIBONACCI QUATERNIONS (59) Q F, + ( - 1 ) 0 /F = 4 v ; ^+r +t * ; ^+t +r 5 T 2W 1 + ( - 1 > r > - - H) + t T r t - ( - i f + r + t T ^ t SECTION 5 I this sectio we give the results coectig Lucas Quaterios T Fiboacci ad Lucas Numbers. The simplest is: (60 > T " i T +l - J T + 2 " k T +3 = 1 5 F +3 (6 «W W = Q 2+2r " ^ " X (62) T F = Q - (-l) + r Q A -r -r 2~2r 0 (63) x ' T +r _,_ F +r _ + T -r F -r = Q 0 2 L OT 2X. - 2(-l) + r Q 0 A (64) T _, F, - T F = F 0 T 0 v ' +r +r -r -r 2r 2 < 65 > T +r F -r = % " ^ " X < 66 > T -rvr = Q 2 + ^ " ^ ( 6 7 ) T + r F - r + T -r F + r = 2 % ' t ' ^ S ^ O < 68 > T + r F - r " T - r F + r = ^ ^ V o (69) T F = Q rt - (-l) Q v ; v ; +r ^2+r ^ r < 70 > T F +r = Q 2 + r + ^ X < 71 > T + r F + T F +r = 2Q 2+r " ^ \ % (72) T ^ F - T F ^ = (~l) +1 F T A +r +r r 0

8 208 SOME RESULTS ON FIBONACCI QUATERNIONS [Apr. (73) W* =Q 2^- ( -A-t ( 7 4 ) T r + t F + r = Q 2 + r + t + ( " 1 ) + r Q?rt So ( 7 5 ) T + r V t + T + t F + r = * W * " ^ + K t % ( 7 6 ) T + r F + t " T + t F + r = ( " ^ V o ( 7 7 ) T + r L - r = T 2 + ^ " ' ^ r ( 7 8 ) VrVr = T 2 + W^r ( 7 9 ) T + r V r + T - r L + r = 2 T 2 + ^"'Vo ( 8 0 ) T + r L - r " T - A + r = ^ ^ V o ( 8 1 ) T + r L = T 2 + r + < " «% (82) T L ^ = T _,_ + (-l) T*. +r 2+r v ' r (83) T, L + T L ^ = 2T 0 _,_ + (-l) L T A +r +r 2+r r 0 ( 8 4 ) T + r L " T V r = ^ ^ r % ( 8 5 ) W + t = T 2 + r + t + ^ ^ r - t ( 8 6 ) V t L + r = T 2 + r + t + ^""Vt ( 8 7 ) T + r V t + T + t L + r = 2 T 2 + r + t + ^ " V l ^ O ( 8 8 ) T + r L + t " T + t V r = ^ ^ H - t S i89) T + r L - r + ^X-A+r = V 1 + <-»*> + ^ " ' ^ i T < - «X

9 ] SOME RESULTS ON FIBONACCI QUATERNIONS 209 < 90) T + r F - r + ^ V ^ = I [ Q 2 ( 1 + ^ " ^'\r (91 > T + r F + t + ( - 1 ) r V + r = I [ S + r + t ( 1 + ( - 1 ) r ) - (-D +t Q r _ t + (-D +r+t Q*_ t ] < 92 > T + r L + t + ^ \ + t \ + v = T 2 a + r t ( 1 + { ~ lf) + ^ V t SECTION 6 Lastly, i this sectio we obtai the iter-relatios betwee the Fiboacci ad Lucas Quaterios (93) Q L + T F = 2Q 2 (94) Q L - T F = 2(-l) Q 0 (95) Q + T = 2Q _,, ' +1 (96) T - Q = 2Q, ' -1 (97) T + Q = 6 [ 2 F Q " 3 F ] + 4 ^ % (98) T " Q = 4 [ 2 F Q - 3 F ( - 1 ) T o ] (99) T Q + T ^ Q ^ = 2T 2 _ 1-15F (100) T Q - T ^ Q ^ = 2 Q 2 _ 1-3 L M-lf(Q Q -3 k ) ( ) T Q + l " T + l Q = 2 ( " 1 ) [ 2 Q l " 3 k ] < 102 > T + A + s " T + S Q + r = ^ - U ^ 1 W o

10 210 SOME RESULTS ON FIBONACCI QUATERNIONS Apr REFERENCES 1. A. F. Horadam, "Complex Fiboacci Numbers ad Fiboacci Quaterios,,! Amer, Math. Mothly, 70, 1963, pp , A. F. Horadam, "A Geeralized Fiboacci Sequece, ff Amer. Math. Mothly, 68, 1961, pp Muthulakshmi R. Iyer, "Idetities Ivolvig Geeralized Fiboacci Numb e r s, " the Fiboacci Quarterly, Vol. 7, No. 1 (Feb. 1969), pp E. Lucas, Theorie des Numbers, Paris, (Cotiued from p. 200.) * SOLUTIONS TO PROBLEMS 1. For ay modulus m, there are m possible residues ( 0, 1, 2, ",m - 1). Successive pairs may come i m 2 ways. Two successive residues determie all residues thereafter. Now i a ifiite sequece of residues there is boud to be repetitio ad hece periodicity. Sice m divides T 0, it must by reaso of periodicity divide a ifiity of members of the sequece. 2. = mk» where m ad k are odd. V ca be writte which is divisible by V = r + s. J _ m _ T7, ivk,, m x k V = (r ) + (s ), 3 r = 2 + 2i >s/2, s = 2-2i *s/2. \ {^y*(i^y 4. The auxiliary equatio is (x - I) 2 = 0, so that T has the form T = Ax l + B x l = A + B. 5. T = 2 / b - 2a\ ^ 4a - b (Cotiued o p. 224.) *

SOME TRIBONACCI IDENTITIES

Mathematics Today Vol.7(Dec-011) 1-9 ISSN 0976-38 Abstract: SOME TRIBONACCI IDENTITIES Shah Devbhadra V. Sir P.T.Sarvajaik College of Sciece, Athwalies, Surat 395001. e-mail : drdvshah@yahoo.com The sequece