On Generalized Fibonacci Quaternions and Fibonacci-Narayana Quaternions. Cristina Flaut & Vitalii Shpakivskyi. Advances in Applied Clifford Algebras

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1 O Geeralized Fiboacci Quaterios ad Fiboacci-Narayaa Quaterios Cristia Flaut & Vitalii Shpakivskyi Advaces i Applied Clifford Algebras ISSN Volume 3 Number 3 Adv. Appl. Clifford Algebras 013) 3: DOI /s

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3 Adv. Appl. Clifford Algebras 3 013), Spriger Basel / published olie May 9, 013 DOI /s Advaces i Applied Clifford Algebras O Geeralized Fiboacci Quaterios ad Fiboacci-Narayaa Quaterios Cristia Flaut* ad Vitalii Shpakivskyi Abstract. I this paper, we ivestigate some properties of geeralized Fiboacci quaterios ad Fiboacci-Narayaa quaterios i a geeralized quaterio algebra. Keywords. Fiboacci quaterios, geeralized Fiboacci quaterios, Fiboacci-Narayaa quaterios. 0. Itroductio The Fiboacci umbers was itroduced by Leoardo of Pisa ) i his book Liber abbaci, book published i 10 AD see [9], p. 1, 3). These umbers were used as a model for ivestigate the growth of rabbit populatios see [4]). The Lati ame of Leoardo was Leoardus Pisaus, also called Leoardus filius Boaccii, shortlyfiboacci. This ame is attached to the followig sequece of umbers 0, 1, 1,, 3, 5, 8, 13, 1,..., with the th term give by the formula: f = f 1 + f,, where f 0 =0,f 1 =1. Fiboacci umbers was kow i Idia before Leoardo s time ad used by the Idia authorities o metrical scieces see [10], p. 30). These umbers have may properties which were studied by may authors see [6], [], [10], [9]). Narayaa was a outstadig Idia mathematicia of the XIV cetury. From him came to us the mauscript Bidzhahaity icomplete), writte i the middle of the XIV cetury. For Narayaa was iterestig summatio *Correspodig author.

4 674 C. Flaut ad V. Shpakivskyi Adv. Appl. Clifford Algebras of arithmetic series ad magic squares. I the middle of the XIV cetury he proved a more geeral summatio. Usig the followig sums = S 1), S 1) 1 + S 1) S 1) = S ), S ) 1 + S ) S ) = S ),..., Narayaa calculated that S m) +1) +)... + m) =. *) m +1) Narayaa applied its rules to the problem of a herd of cows ad heifers see [16], [1], [13], [1]). Narayaa problem [1]). A cow aually brigs heifers. Every heifer, begiig from the fourth year of his life also brigs heifer. How may cows ad calves will be after 0 years? Narayaa s calculatio is i the followig: 1) a cow withi 0 years brigs 0 heifers of the first geeratio; ) the first heifer of the first geeratio brigs 17 heifers secod geeratio, the secod heifer of the first geeratio brigs 16 heifers secod geeratio etc. The total i the secod geeratio will be = S 1) 17 cows ad calves; 3) the first heifer of the sevetee heifers of the secod geeratio brigs 14 heifers of the third geeratio, the secod heifer of the sevetee heifers of the secod geeratio brigs 13 heifers of third geeratio, etc. The total heifers of the first geeratio brigs = S 1) 13 heads. Now, all heifers of the secod geeratio brigs S 1) 14 +S1) S1) heads i the third geeratio. S ) 14 Similarly, Narayaa calculated total umber i the herd after 0 years: =1+0+S 1) 17 + S) S6). Usig formula *), he obtaied: = = = This problem ca be solved i the same way that Fiboacci solved its problem about rabbits see [8], [9], [1], [13]). I the begiig of the first year were 1 cow ad 1 heifer which bor. We have ow heads. I the begiig of the secod year ad i the begiig of the third year the umber of heads icreased by oe. Therefore the umber of heads are 3 ad 4, respectively. From the fourth year, the umber of heads i the herd is defied by recurrece formulae: x 4 = x 3 + x 1,x 5 = x 4 + x,...,x = x 1 + x 3,

5 Vol ) O Geeralized Fiboacci Quaterios 675 sice the umber of cows for ay year is equal with the umber of cows of the previous year plus the umber of heifers which was bor = umber of heads that were three years ago) see [1]). We have the sequece, 3, 4, 6, 9,..., u +1 = u + u. Computig, we obtai that u 0 = 745 see [8], [9], [1], [13], [1]). Now, we ca cosider the sequece 1, 1, 1,, 3, 4, 6, 9,..., u +1 = u + u, with,u 0 =0,u 1 =1,u =1. These umbers are called the Fiboacci- Narayaa umbers see [3]). Ithesamepaper[Di,St; 03], authors proved some basic properties of Fiboacci-Narayaa umbers, amely: 1) u 1 + u u = u ) u 1 + u 4 + u u 3 = u ) u + u 5 + u u 3 1 = u 3. 4) u 3 + u 6 + u u 3 = u ) u +m = u 1 u m+ + u u m + u 3 u m+1. 6) u = u +1 + u 1 u. 7) If i the sequeces {u },=7k +4,=7k +6,=7k, whe k = 0, 1,,...,theu is eve. 8) If i the sequeces {u } = 8k, = 8k 1, = 8k 3, whe k =0, 1,,...,the3 u. Aother property of Fiboacci-Narayaa umbers was proved i [14]. For all atural, we have u = [/3] m [/3] u [/3] m, where [a] isaitegerpartofa ad k! = k! k)!,k!= k, k N. Let H β 1,β ) be the geeralized real quaterio algebra, the algebra oftheelemetsoftheforma = a 1 1+a e + a 3 e 3 + a 4 e 4, where a i R,i {1,, 3, 4}, ad the basis elemets {1,e,e 3,e 4 } satisfy the followig multiplicatio table: 1 e e 3 e e e 3 e 4 e e β 1 e 4 β 1 e 3 e 3 e 3 e 4 β β e e 4 e 4 β 1 e 3 β e β 1 β We deote by t a) ad a) the trace ad the orm of a real quaterio a. The orm of a geeralized quaterio has the followig expressio a) = a 1 + β 1 a + β a 3 + β 1 β a 4. For β 1 = β =1, we obtai the real divisio algebra H.

6 676 C. Flaut ad V. Shpakivskyi Adv. Appl. Clifford Algebras 1. Prelimiaries I the preset days, several mathematicias studied properties of the Fiboacci sequece. I [6], the author geeralized Fiboacci umbers ad gave may properties of them: h = h 1 + h,, where h 0 = p, h 1 = q, with p, q beig arbitrary itegers. I the same paper [6], relatio 7), the followig relatio betwee Fiboacci umbers ad geeralized Fiboacci umbers was obtaied: h +1 = pf + qf ) The same author, i [7], defied ad studied Fiboacci quaterios ad geeralized Fiboacci quaterios i the real divisio quaterio algebra ad foud a lot of properties of them. For the geeralized real quaterio algebra, the Fiboacci quaterios ad geeralized Fiboacci quaterios are defied i the same way: F = f 1+f +1 e + f + e 3 + f +3 e 4, for the th Fiboacci quaterios, ad H = h 1+h +1 e + h + e 3 + h +3 e 4, for the th geeralized Fiboacci quaterios. I the same paper, we fid the orm formula for the th Fiboacci quaterios: F )=F F =3f +3, 1.) where F = f 1 f +1 e f + e 3 f +3 e 4 is the cojugate of the F i the algebra H. After that, may authors studied Fiboacci ad geeralized Fiboacci quaterios i the real divisio quaterio algebra givig more ad surprisig ew properties for example, see [15], [11] ad [5]). M. N. S. Swamy, i [15], formula 17), obtaied the orm formula for the th geeralized Fiboacci quaterios: H ) = H H = 3pq p )f + +p + q )f +3, where H = h 1 h +1 e h + e 3 h +3 e 4 is the cojugate of the H i the algebra H. Similar to A. F. Horadam, we defie the Fiboacci-Narayaa quaterios as U = u 1+u +1 e + u + e 3 + u +3 e 4, where u are the th Fiboacci-Narayaa umber. I this paper, we give some properties of geeralized Fiboacci quaterios ad Fiboacci-Narayaa quaterios.

7 Vol ) O Geeralized Fiboacci Quaterios 677. Geeralized Fiboacci Quaterios As i the case of Fiboacci umbers, umerous results betwee Fiboacci geeralized umbers ca be deduced. I the followig, we will study some properties of the geeralized Fiboacci quaterios i the geeralized real quaterio algebra H β 1,β ). Let F = f 1+f +1 e + f + e 3 + f +3 e 4 be the th Fiboacci quaterio ad H = h 1+h +1 e + h + e 3 + h +3 e 4 be the th geeralized Fiboacci quaterio. A first questio which ca arise is what algebraic structure have these elemets? The aswer will be foud i the below theorem, deotig first a th geeralized Fiboacci umber ad a th geeralized Fiboacci elemet with h p,q we emphasis the startig itegers p ad q., respectively H p,q. I this way, Theorem.1. The set H = {H p,q /p,q Z} {0} is a Z-module. Proof. Ideed, ah p,q + bh p,q = H ap+bp,aq+bq H, where a, b, p, q, p,q Z. Theorem.. i) For the Fiboacci quaterio elemets, we have 1) m+1 F m = 1) +1 F 1 +1+e 3 + e 4..1) ii) For the geeralized Fiboacci quaterio elemets, the followig relatio is true, 1) m+1 Hm p,q = 1) +1 H p,q 1 p+q+pe +qe 3 +pe 4 +qe 4..) Proof. i) From [], we kow that 1) m+1 f m = 1) +1 f ) It results: 1) m+1 F m = 1) m+1 f m + e + e 3 1) m+1 f m+ + e 4 1) m+1 f m+1 1) m+1 f m+3 = 1) +1 f 1 +1 e [ 1) +1 f 1 + 1) + f +1 ] [ ] + e 3 1) +1 f ) f ) +1 f + e 4 [ 1) +1 f ) + f ) +3 f + + 1) +4 [ f +3 ] = 1) +1 f ) +1 e f + e 3 1) +1 f ) +1]

8 678 C. Flaut ad V. Shpakivskyi Adv. Appl. Clifford Algebras e 4 1) +1 [ f + 1) +1] = 1) +1 f 1 + f e + f +1 e 3 + f + e 4 )+1+e 3 + e 4 = 1) +1 F 1 +1+e 3 + e 4. ii) Usig relatios 1.1) ad.3),we have = = 1) m+1 H p,q m 1) m+1 h p,q m + e + e 3 1) m+1 h p,q m+ + e 4 1) m+1 pf m 1 + 1) m+1 h p,q m+1 1) m+1 h p,q m+3 1) m+1 qf m + e 1) m+1 pf m + e + e 3 1) m+1 pf m+1 + e 3 + e 4 1) m+1 pf m+ + e 4 1) m+1 qf m+1 1) m+1 qf m+ 1) m+1 qf m+3 = p 1) +1 f p + q 1) +1 f 1 + q ] + e p 1) +1 f 1 + pe + e q [ 1) +1 f +1 1) +1 f 1 ] + e 3 p [ 1) +1 f +1 1) +1 f 1 ] + e 3 q [ 1) +1 f ) f ) +1 f + ] + e 4 p [ 1) +1 f ) f ) +1 f + e 4 q[ 1) +1 f ) + f ) +3 f + + 1) +4 f +3 ] = p 1) +1 f p + q 1) +1 f 1 + q + e p 1) +1 f 1 + pe + e q 1) +1 f + e 3 p 1) +1 f + e 3 q 1) +1 [f 1 + 1) +1 f +1 + f + ] + e 4 p 1) +1 [ f 1 + 1) +1 f +1 + f + ] e 4 q 1) +1 [ f 1 1) +1 f +1 + f + f +3 ] = p 1) +1 f p + q 1) +1 f 1 + q

9 Vol ) O Geeralized Fiboacci Quaterios e p 1) +1 f 1 + pe + e q 1) +1 f + e 3 p 1) +1 f [ + e 3 q 1) +1 f ) +1] + e 4 p 1) +1 [ 1) +1 + f +1 ] [ e 4 q 1) +1 f + 1) +1] = 1) +1 H p,q 1 p + q + pe + qe 3 + pe 4 + qe 4. From the above Theorem, we ca remark that all idetities valid for the Fiboacci quaterios ca be easy adapted i a approximative similar expressio for the geeralized Fiboacci quaterios, if we use relatio 1.1), a true relatio i the both algebras H β 1,β )adh. Propositio.3. If h +1 = pf + qf +1 =0, the we have: H+1 =3 q [ ] f f +1 f +1 f f +,.4) where H+1 H β 1,β ). Proof. Sice h +1 =0, it results that t H +1 )=h +1 =0, therefore H +1 )=H+1. Fromh = pf + qf +1 =0,wehavep = qf+1 f ad we obtai: ad p +pq = q f +1 f p + q = q f+1 f sice f+1 + f = f +1. It results q f +1 f + q =q f +1 + f f = q f +1 f f = q f +1 f, H +1 ) = 3[p +pq)f + +p + q )f +1 ] =3 q f [ f +1 f f + + f+1]. I the followig, we will compute the orm of a Fiboacci quaterio ad of a geeralized Fiboacci quaterio i the algebra H β 1,β ). Let F = f 1+f +1 e +f + e 3 +f +3 e 4 be the th Fiboacci quaterio, the its orm is F )=f + β 1 f +1 + β f + + β 1 β f +3. Usig recurrece of Fiboacci umbers ad relatios f + f 1 = f 1, N,.5) f = f +f f 1, N,.6)

10 680 C. Flaut ad V. Shpakivskyi Adv. Appl. Clifford Algebras from [], we have F )=f + β 1 f+1 + β f+ + β 1 β f+3 = f + β 1 f+1 + β f + + β 1 f+3 ) = f +1 +β 1 1) f+1 + β f+5 +β 1 1)f+3 ) = f +1 + β f +5 +β 1 1) f+1 + β f+3 ) =1+β ) f +1 +3β f + +β 1 1) f+1 + β f+3 ) + +β 1 1) f+1 + β f+3 ) + +β 1 1) f + f f +1 + β f +6 β f + f +3 ) + +β 1 1)[f + + β f +6 f f +1 + β f + f +3 )] + +β 1 1)[f + + β f +6 f f +1 +β f++β ) f +1 f + ] + +β 1 1)[f + + β f +6 f f +1 +β f++βf +1+β ) f f β 1 1)[f + + β f +6 1+β ) f f +1 β f +3 ] + +β 1 1)[f + + β f +4 + β f +3 + β f +4 β f +3 1+β ) f f +1 ] + +β 1 1)[f + +β f +4 β f +3 1+β ) f f +1 ] + +β 1 1)[f + +β f + +β f +3 β f +3 1+β ) f f +1 ] + +β 1 1)[1 + β ) f + + β f +3 1+β ) f f +1 ] + +β 1 1)[h 1+β,β +3 1+β ) f f +1 ] + +β 1 1)h 1+β,β +3 β 1 1) 1+β ) f f +1. We just proved Theorem.4. The orm of the th Fiboacci quaterio F i a geeralized quaterio algebra is F ) =h 1+β,3β + +β 1-1)h 1+β,β +3 -β 1-1) 1+β ) f f +1..7) Usig formula.7) ad relatio 1.1) whe β 1 = β =1, we obtai formula 1.). Usig the above theorem ad relatios.5) ad.6), we ca compute the orm of a geeralized Fiboacci quaterio i a geeralized quaterio algebra. Let H = h 1+h +1 e + h + e 3 + h +3 e 4 be the th geeralized Fiboacci quaterio. The orm is

11 Vol ) O Geeralized Fiboacci Quaterios 681 H p,q )=h + β 1 h +1 + β h + + β 1 β h +3 =pf 1 +qf ) + β 1 pf +qf +1 ) + β pf +1 +qf + ) + β 1 β pf + +qf +3 ) = p f 1 + β 1 f + β f+1 + β 1 β f+ ) + q f + β 1 f+1 + β f+ + β 1 β f+3 ) +pq f 1 f + β 1 f f +1 + β f +1 f + + β 1 β f +3 f + ) = p h 1+β,3β + p β 1 1)h 1+β,β +1 p β 1 1) 1+β ) f 1 f + q h 1+β,3β + + q β 1 1)h 1+β,β +3 q β 1 1) 1+β ) f f +1 +pq 1-β 1 ) f f 1 +pqβ 1 f +pqβ 1-β 1 ) f +1 f + +pqβ 1 β f +4 = p h 1+β,3β + p β 1 1)h 1+β,β +1 + q h 1+β,3β + + q β 1 1)h 1+β,β +3 p β 1 1) pβ + p + q) f 1 f q β 1 1) 1+β ) f f +1 + h pqβ1,pqβ1β +1 +pqβ 1 β f + f +3 )+pqβ 1 β 1 ) f +1 f +. From the above, we proved Theorem.5. The orm of the th geeralized Fiboacci quaterio H p,q i a geeralized quaterio algebra is H p,q )=p h 1+β,3β + p β 1 1)h 1+β,β +1 + q h 1+β,3β + + q β 1 1)h 1+β,β +3 p β 1 1) pβ + p + q) f 1 f q β 1 1) 1+β ) f f +1 + h pqβ1,pqβ1β +1 +pqβ 1 β f + f +3 )+pqβ 1 β 1 ) f +1 f +..8) It is kow that the expressio for the th term of a Fiboacci elemet is f = 1 [α β ]= α [1 β ],.9) 5 5 α where α = 1+ 5 ad β = 1 5. From the above, we ca compute the followig, lim F ) = lim f + β 1 f+1 + β f+ + β 1 β f+3) where = lim α 5 +β 1 =sgeβ 1,β ) α + 5 α +4 α +6 +β +β 1 β ) 5 5 Eβ 1,β )= β 1 5 α + β 5 α4 + β 1β 5 α6 ) = β 1 α +1)+β 3α +)+β 1 β 8α +5)) = 1 5 [1 + β 1 +β +5β 1 β + α β 1 +3β +8β 1 β )],

12 68 C. Flaut ad V. Shpakivskyi Adv. Appl. Clifford Algebras sice α = α +1. If Eβ 1,β ) > 0, there exist a umber 1 N such that for all 1 we have h 1+β,3β + +β 1 1)h 1+β,β +3 β 1 1) 1 + β ) f f +1 > 0. Ithesameway,ifEβ 1,β ) < 0, there exist a umber N such that for all we have h 1+β,3β + +β 1 1)h 1+β,β +3 β 1 1) 1 + β ) f f +1 < 0. Therefore for all β 1,β R with Eβ 1,β ) 0, i the algebra H β 1,β ) there is a atural umber 0 =max{ 1, } such that F ) 0, hece F is a ivertible elemet for all 0. Usig the same argumets, we ca compute lim Hp,q )) = lim h + β 1 h +1 + β h + + β 1 β h ) +3 = lim [pf 1 + qf ) + β 1 pf + qf +1 ) + β pf +1 + qf + ) + β 1 β pf + + qf +3 ) ] =sge β 1,β ) where E β 1,β ) = 1 5 [p + αq) + β 1 pα + α q ) + β pα + α 3 q ) + β1 β pα 3 + α 4 q ) ] = 1 5 p + αq) [1 + β 1 α + β α 4 + β 1 β α 6 ] = 1 5 p + αq) Eβ 1,β ). Therefore for all β 1,β R with E β 1,β ) 0 i the algebra H β 1,β ) there exist a atural umber 0 such that H p,q ) 0, hece H p,q is a ivertible elemet for all 0. Now, we proved Theorem.6. For all β 1,β R with E β 1,β ) 0, there exists a atural umber such that for all Fiboacci elemets F ad geeralized Fiboacci elemets H p,q are ivertible elemets i the algebra H β 1,β ). Remark.7. Algebra H β 1,β ) is ot always a divisio algebra, ad sometimes ca be difficult to fid a example of ivertible elemet. Above Theorem provides us ifiite sets of ivertible elemets i this algebra, amely Fiboacci elemets ad geeralized Fiboacci elemets. 3. Fiboacci-Narayaa Quaterios I this sectio, we will study some properties of Fiboacci-Narayaa elemets i the algebra H β 1,β ).

13 Vol ) O Geeralized Fiboacci Quaterios 683 Theorem 3.1. For the Fiboacci-Narayaa quaterio U, we have a) b) U m = U +3 U, U 3m = U e 4. Proof. a) U m = +1 u m + e + u m + e 3 Sice u 0 = 0, we cosider that the term m= +3 u m + e 4 m=3 u m is equal with ca use property 1) from the itroductio ad we obtai u m =: ). ) =u +3 1+e u +4 1) + e 3 u +5 ) + e 4 u +6 3) = U e +e 3 +3e 4 )=U +3 U. b) Sice u 0 =0, the term U 3m = u 3m + e u 3m+1 + e 3 u 3m is equal with usig properties 4), ), 3), ad agai 4), we have u 3m+ + e 4 u m. We u 3m, therefore u 3m+3 =: ); ) =u u 3+ e +u 3+3 e 3 +u 3+4 1)e 4 = U e 4. Let {u } be a Fiboacci-Narayaa sequece, ad let U = u 1+ u +1 e + u + e 3 + u +3 e 4 be the th Fiboacci-Narayaa quaterio. The fuctio fx) =a 0 + a 1 x + a x a x +... is called the geeratig fuctio for the sequece {a 0,a 1,a,...}. I [5], the author foud a geeratig fuctio for Fiboacci quaterios. I the followig theorem, we established the geeratig fuctio for Fiboacci-Narayaa quaterios. Theorem 3.. The geeratig fuctio for the Fiboacci-Narayaa quaterio U is Gt)= U 0+U 1 -U 0 )t+u -U 1 )t 1 t t 3 = e 1+e +e 3 +1+e 3 )t +e +e 3 )t 1 t t ) Proof. Assumig that the geeratig fuctio of the quaterio Fiboacci- Narayaa sequece {U } has the form Gt) = U t, we obtai that =0

14 684 C. Flaut ad V. Shpakivskyi Adv. Appl. Clifford Algebras U t t U t t 3 =0 =0 =0 U t = U 0 + U 1 t + U t + U 3 t U 0 t U 1 t U t 3 U 3 t 4... U 0 t 3 U 1 t 4 U t 5 U 3 t 6... = U 0 +U 1 U 0 )t +U U 1 )t, sice U = U 1 + U 3, 3adthecoefficietsoft for 3areequal with zero. It results U 0 +U 1 U 0 )t+u U 1 )t = U t 1 t t 3 ), =0 or i equivalet form U 0 +U 1 U 0 )t+u U 1 )t 1 t t 3 = U t. =0 The theorem is proved. Theorem 3.3 Biet-Cauchy formula for Fiboacci-Narayaa umbers). Let u = u 1 + u 3, 3 be the th Fiboacci-Narayaa umber, the 1 [ u = α +1 γ-β) +β +1 α-γ) +γ +1 β-α) ], 3.) α-β)β-γ)γ-α) where α, β, γ are the solutios of the equatio t 3 t 1=0. Proof. Supposig that u = Aα + Bβ + Cγ,A,B,C C ad usig the recurrece formula for the Fiboacci-Narayaa umbers, u = u 1 + u 3, it results that α, β, γ are the solutios of the equatio t 3 t 1=0. Sice u 0 =0,u 1 =1,u =1, we obtai the followig system: A + B + C =0, Aα + Bβ + Cγ =1, Aα + Bβ + Cγ =1. 3.3) The determiat of this system is a Vadermode determiat ad ca be computed easily. It is Δ = α-β)β-γ)γ-α) 0. Usig Cramer s rule, the solutios of the system 3.3) are A = B = α γ β) α β)β γ)γ α) = α β α)γ α), β α γ) α β)β γ)γ α) = β α β)γ β), γ β α) C = α β)β γ)γ α) = γ β γ)α γ), therefore relatio 3.) is true.

15 Vol ) O Geeralized Fiboacci Quaterios 685 Theorem 3.4 Biet-Cauchy formula for the Fiboacci-Narayaa quaterios). Let U = u 1+u +1 e + u + e 3 + u +3 e 4 be the th Fiboacci- Narayaa quaterio, the U =D α+1 β-α)γ-α) +E β +1 α-β)γ-β) +F γ +1 β-γ)α-γ), 3.4) where α, β, γ are the solutios of the equatio t 3 t 1=0ad D =1+αe 1 + α e + α 3 e 3, E =1+βe 1 + β e + β 3 e 3, F =1+γe 1 + γ e + γ 3 e 3. Proof. Usig relatio 3.), we have that U = u 1+u +1 e + u + e 3 + u +3 e 4 1 = α-β)β-γ)γ-α) [ α +1 γ-β)+β +1 α-γ)+γ +1 β-α) ) 1 + α + γ β)+β + α γ)+γ + β α) ) e 1 + α +3 γ β)+β +3 α γ)+γ +3 β α) ) e + α +4 γ β)+β +4 α γ)+γ +4 β α) ) e 3 ] 1 = α β)β γ)γ α) [α+1 γ-β) 1+αe 1 +α e +α 3 ) e 3 + β +1 α-γ) 1+βe 1 + β e + β 3 ) e 3 + γ +1 β-α) 1+γe 1 + γ e + γ 3 ) e 3 ]. For egative, theth Fiboacci-Narayaa umber is defied as i the followig: u = u +3 u +, u 0 =0,u 1 =1,u =1.Ithesamewayis defied the Fiboacci-Narayaa quaterio U for egative. Theorem 3.5. Let U = u 1+u +1 e +u + e 3 +u +3 e 4 be the th Fiboacci- Narayaa quaterio, therefore the followig relatios are true: 1) i U i 1 = U 3 1. ) i=0 i U 3 i 1 = U 4 1. i=0 Proof. 1) Usig Newto s formula, it results that t +1 ) = 0 t ) + 1 t ) 1 + t ) = 0 t + 1 t + t

16 686 C. Flaut ad V. Shpakivskyi Adv. Appl. Clifford Algebras From here, we have that i U i 1 i=0 = 0 U U 3 + U U 1 = 0 α D β α)γ α) + E β α β)γ β) + F γ ) β γ)α γ) + 1 α D β α)γ α) + E β α β)γ β) + F γ ) +... β γ)α γ) ) + 1 D β α)γ α) + E 1 α β)γ β) + F 1 β γ)α γ) 1 = D β α)γ α) 0 α + 1 α ) 1 + E α β)γ β) 0 β + 1 β ) 1 + F β γ)α γ) 0 γ + 1 γ ) 1 = D α +1 ) 1 + E β +1 ) β α)γ α) α β)γ β) 1 + F γ +1 ) β γ)α γ) = D β α)γ α) α3 + E α β)γ β) β3 + F β γ)α γ) γ3 = U 3 1. We used that α 3 = α +1,β 3 = β +1,γ 3 = γ +1. ) Sice t 3 = t + 1, startig from relatio t 3 + t ) = t 4,for t {α, β, γ}, by straightforward calculatios as i ), we obtai the asked relatio. Coclusios I this paper we ivestigated some ew properties of geeralized Fiboacci quaterios ad Fiboacci-Narayaa quaterios. Sice Fiboacci-Narayaa quaterios was ot itesive studied util ow, we expect to fid i the future more ad surprisig ew properties. We studied these elemets for the beauty of the relatios obtaied, but the mai reaso is that the elemets of this type, amely Fiboacci X elemets, where X {quaterios, geeralized quaterios}, ca provide us may importat iformatio i the algebra H β 1,β ), as for example: sets of ivertible elemets i algebraic structures without divisio. Ackowledgemets. Authors thak referee for his/her patiece ad suggestios which help us to improve this paper.

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18 688 C. Flaut ad V. Shpakivskyi Adv. Appl. Clifford Algebras Vitalii Shpakivskyi Departmet of Complex Aalysis ad Potetial Theory Istitute of Mathematics of the Natioal Academy of Scieces of Ukraie 3, Tereshchekivs ka st Kiev-4 Ukraie complex/ Received: September 4, 01. Accepted: December 7, 01.