Europea Joural of Combiatorics 9 (008) 70 7 wwwelseviercom/locate/ec The Biet formula, sums ad represetatios of geeralized Fiboacci p-umbers Emrah Kilic TOBB ETU Uiversity of Ecoomics ad Techology, Mathematics Departmet, 06560 Sogutozu, Akara, Turkey Received 0 Jue 006; accepted 6 March 007 Available olie April 007 Abstract I this paper, we cosider the geeralized Fiboacci p-umbers ad the we give the geeralized Biet formula, sums, combiatorial represetatios ad geeratig fuctio of the geeralized Fiboacci p-umbers Also, usig matrix methods, we derive a explicit formula for the sums of the geeralized Fiboacci p-umbers c 007 Elsevier Ltd All rights reserved Itroductio We cosider a geeralizatio of well-kow Fiboacci umbers, which are called Fiboacci p-umbers The Fiboacci p-umbers F p () are defied by the followig equatio for > p+ F p () = F p ( ) + F p ( p ) () with iitial coditios F p () = F p () = = F p (p) = F p (p + ) = If we take p =, the the sequece of Fiboacci p-umbers, {F p ()}, is reduced to the well-kow Fiboacci sequece {F } The Fiboacci p-umbers ad their properties have bee studied by some authors (for more details see [,4 6,8,3 6,9]) E-mail address: ekilic@etuedutr 095-6698/$ - see frot matter c 007 Elsevier Ltd All rights reserved doi:006/ec00703004
70 E Kilic / Europea Joural of Combiatorics 9 (008) 70 7 I 843, Biet gave a formula which is called Biet formula for the usual Fiboacci umbers F by usig the roots of the characteristic equatio x x = 0 : α = + 5, β = 5 F = α β α β where α is called Golde Proportio, α = + 5 (for details see [7,30,8]) I [], Levesque gave a Biet formula for the Fiboacci sequece by usig a geeratig fuctio I [], the authors cosidered a compaio matrix ad its th power, the gave the combiatorial represetatio of the sequece geerated by the th power the matrix Further i [5], the authors derived aalytical formulas for the Fiboacci p-umbers ad the showed these formulas are similar to the Biet formulas for the classical Fiboacci umbers Also, i [], the authors gave the geeralized Biet formulas ad the combiatorial represetatios for the geeralized orderk Fiboacci [3] ad Lucas [7] umbers I [0], the authors defied the geeralized order-k Pell umbers ad gave the Biet formula for the geeralized Pell sequece For the commo geeralizatio of the geeralized order-k Fiboacci ad Pell umbers, ad its geeratig matrix, sums ad combiatorial represetatio, we refer readers to [9] I this paper, we cosider the geeralized Fiboacci p-umbers ad give the geeralized Biet formula, combiatorial represetatios ad sums of the geeralized Fiboacci p-umbers by usig the matrix method The geeratig matrix for the geeralized Fiboacci p-umbers is give by Stakhov [3] as follows: Let Q p be the followig (p + ) (p + ) compaio matrix : 0 0 0 0 0 0 0 0 0 0 0 Q p = 0 0 0 0 0 0 0 0 ad the th power of the matrix Q p is Q p = F p ( + ) F p ( p + ) F p ( ) F p () F p () F p ( p) F p ( ) F p ( ) F p ( p + ) F p ( p + ) F p ( p) F p ( p + ) F p ( p + ) F p ( p + ) F p ( p ) F p ( p) The matrix Q p is said to be a geeralized Fiboacci p-matrix The geeralized Biet formula I this sectio, we give the geeralized Biet formula for the geeralized Fiboacci p- umbers We start with the followig results Lemma Let a p = p p p p The ap > a p+ for p > () (3)
E Kilic / Europea Joural of Combiatorics 9 (008) 70 7 703 Proof Sice p 3 p > 0 ad p >, ( p + p + ) ( p ) > p 4 Thus, p > p p p+ Therefore, for p >, p p ( ) p p > p p+ ad so p p+ p p > ( p p+ ) p ( The we have p p > p p+ p+) So the proof is easily see Lemma The characteristic equatio of the Fiboacci p-umbers x p x p = 0 does ot have multiple roots for p > Proof Let f (z) = z p z p Suppose that α is a multiple root of f (z) = 0 Note that α 0 ad α Sice α is a multiple root, f (α) = α p α p = 0 ad f (α) = pα p (p ) α p = 0 The f (α) = α p (pα (p )) = 0 Thus α = p p, ad hece 0 = f (α) = α p + α p + = α p ( α) + p p ( = p ) + = p p + p p p p = a p + Sice, by Lemma, a = 4 < ad a p > a p+ for p >, a p, which is a cotradictio Therefore, the equatio f (z) = 0 does ot have multiple roots We suppose that f (λ) is the characteristic polyomial of the geeralized Fiboacci p-matrix Q p The, f (λ) = λ p+ λ p, which is a well-kow fact from the compaio matrices Let λ, λ,, λ p+ be the eigevalues of the matrix Q p The, by Lemma, we kow that λ, λ,, λ p+ are distict Let Λ be a (p + ) (p + ) Vadermode matrix as follows: λ p λ p λ λ p Λ = λ p λ λ p p+ λ p p+ λ p+ We deote Λ T by V Let dk i = λ +p+ i λ +p+ i λ +p+ i p+ ad V (i) be a (p + ) (p + ) matrix obtaied from V by replacig the th colum of V by d i k The we ca give the geeralized Biet formula for the geeralized Fiboacci p-umbers with the followig theorem
704 E Kilic / Europea Joural of Combiatorics 9 (008) 70 7 Theorem 3 Let F p () be the th geeralized Fiboacci p-umber; the det V (i) q i = det (V ) where Q p = [ ] q i ad qi = F p ( + i p) for ad q i, = F p ( + i) for = Proof Sice the eigevalues of the matrix Q p are distict, the matrix Q p is diagoalizable It is easy to show that Q p V = V D, where D = diag(λ, λ,, λ p+ ) Sice the Vadermode matrix V is ivertible, V Q p V = D Hece, the matrix Q p is similar to the diagoal matrix D So we have the matrix equatio Q p V = V D Sice Q p = [ q i ], we have the followig liear system of equatios: q i λ p + q iλ p + + q i,p+ = λ p++ i q i λ p + q iλ p + + q i,p+ = λ p++ i q i λ p p+ + q iλ p p+ + + q i,p+ = λ p++ i p+ Thus, for each =,,, p +, we obtai det V (i) q i = det (V ) So the proof is complete Thus, we give the Biet formula for the th Fiboacci p-umber F p () by the followig corollary Corollary 4 Let F p () be the th Fiboacci p-umber The det V () det V () p+ F p () = = det (V ) det (V ) Proof The coclusio is immediate result of Theorem 3 by takig i =, = or i =, = p + The followig lemma ca be obtaied from [] Lemma 5 Let the matrix Q p = [ q i ] be as i (3) The q i = (m,,m p+) m + m + + + m p+ m + m + + m p+ m + m + + m p+ m, m,, m p+ where the summatio is over oegative itegers satisfyig m + m + + (p + )m p+ = i +, ad defied to be if = i The we have the followig corollaries
E Kilic / Europea Joural of Combiatorics 9 (008) 70 7 705 Corollary 6 Let F p () be the geeralized Fiboacci p-umber The F p () = (m,,m p+) m p+ m + m + + m p+ m + m + + m p+ m, m,, m p+ where the summatio is over oegative itegers satisfyig m + m + + (p + )m p+ = + p Proof I Lemma 5, whe i = ad = p +, the the coclusio ca be directly see from (3) Corollary 7 Let F p () be the geeralized Fiboacci p-umber The m + m F p () = + + m p+ m, m,, m p+ (m,,m p+) where the summatio is over oegative itegers satisfyig m + m + + (p + )m p+ = Proof I Lemma 5, if we take i = ad =, the we have the corollary from (3) We cosider the geeratig fuctio of the geeralized Fiboacci p-umbers We give the followig lemma Lemma 8 Let F p () be the th geeralized Fiboacci umber, the for > x = F p ( p + )x p + p F p ( p + ) x = Proof We suppose that = p + ; the by the defiitio of the Fiboacci p-umbers x p+ = F p ()x p + F p () = x p + Now we suppose that the equatio holds for ay iteger, > p + The we show that the equatio holds for + Thus, from our assumptio ad the characteristic equatio the Fiboacci p-umbers, ( ) p x + = x x = F p ( p + )x p + F p ( p + ) x x = F p ( p + ) ( x p + ) + = p F p ( p + ) x = = F p ( p + )x p + F p ( p + ) + F p ( p + ) x p + F p ( p + )x p + + F p ( p + )x + F p ( p)x = [ F p ( p + ) + F p ( p + ) ] x p + F p ( p + )x p + F p ( p + 3) x p + + F p ( p)x + F p ( p + ) (4)
706 E Kilic / Europea Joural of Combiatorics 9 (008) 70 7 Usig the defiitio of the geeralized Fiboacci p-umbers, we have F p ( p + ) + F p ( p + ) = F p ( p + ) Therefore, we ca write the Eq (4) as follows x + = F p ( p + )x p + F p ( p + ) x p + F p ( p + 3) x p + + F p ( p)x + F p ( p + ) p = F p ( p + )x p + F p ( p + ) x (5) which is what was desired = Now we give the geeratig fuctio of the geeralized Fiboacci p-umbers: Let The G p (x) = F p () + F p ()x + F p (3)x + + F p ( + )x + G p (x) xg p (x) x p+ G p (x) = ( x x p+) G p (x) By the Eq (5), we have ( x x p+) G p (x) = F p () = Thus G p (x) = ( x x p+) for 0 x + x p+ < Let f p (x) = x + x p+ The, for 0 f p (x) <, we have the followig lemma Lemma 9 For positive itegers t ad, the coefficiet of x i ( f p (x) ) t is t t, p + t =0 where the itegers satisfy p + t = Proof From the above results, we write ( f p (x) ) t = (x + x p+) t = x t ( + x p) t = x t t =0 t x p I the above equatio, we cosider the coefficiet of x For positive itegers t ad such that p + t = ad t, the coefficiets of x are t t, p + t =0 So we have the required coclusio Now we ca give a represetatio for the geeralized Fiboacci p-umbers by the followig theorem
E Kilic / Europea Joural of Combiatorics 9 (008) 70 7 707 Theorem 0 Let F p () be the th geeralized Fiboacci p-umber The, for positive itegers t ad, F p ( + ) = p+ t t =0 t where the itegers satisfy p + t = Proof Sice G p (x) = F p () + F p () x + F p (3)x + + F p ( + ) x + = x x p+ ad f p (x) = x + x p+, the coefficiet of x is the ( + )th geeralized Fiboacci p-umber, F p ( + ) i G p (x) Thus G p (x) = x x p+ = f p (x) = + f p (x) + ( f p (x) ) ( + + f p (x) ) + = + x ( + x p) + x x p + + x =0 =0 x p + As we eed the coefficiet of x, we oly cosider the first + terms o the right-side Thus by Lemma 9, the proof is complete Now we give a expoetial represetatio for the geeralized Fiboacci p-umbers [ l G p (x) = l (x + x p+)] [ = l (x + x p+)] [ ( = x + x p+) ( x + x p+) (x + x p+) ] [ ( = x + x p ) + ( + x p ) ( + + + x p ) ] + ( = x + x p ) Thus, G p (x) = exp =0 ( x =0 ( + x p ) )
708 E Kilic / Europea Joural of Combiatorics 9 (008) 70 7 3 Sums of the geeralized Fiboacci p-umbers by matrix methods I this sectio, we defie a (p + ) (p + ) matrix T, ad the we show that the sums of the geeralized Fiboacci p-umbers ca be obtaied from the th power of the matrix T Defiitio For p, let T = ( t i ) deote the (p + ) (p + ) matrix byt = t = t = t,p+ =, t i+,i = for i p + ad 0 otherwise Clearly, by the defiitio of the matrix Q p, 0 0 0 0 0 0 0 0 0 0 T = 0 0 0 0 0 0 0 0 0 0 or T = 0 Q p 0 where the (p + ) (p + ) matrix Q p give by () Let S deote the sums of the geeralized Fiboacci p-umbers from to, that is: S = F p (i) i= Now we defie a (p + ) (p + ) matrix C as follows 0 0 S C = S Q p S p where Q p give by (3) The we have the followig theorem Theorem Let the (p + ) (p + ) matrices T ad C be as i (6) ad (8), respectively The, for : C = T Proof We will use the iductio method to prove that C = T If =, the, by the defiitio of the matrix C ad geeralized Fiboacci p-umbers, we have C = T Now we suppose that the equatio holds for The we show that the equatio holds for + Thus, T + = T T ad by our assumptio, T + = C T (6) (7) (8)
E Kilic / Europea Joural of Combiatorics 9 (008) 70 7 709 Sice S + = S + F p ( + ) ad usig the defiitio of the geeralized Fiboacci umbers, we ca derive the followig matrix recurrece relatio C T = C + So the proof is complete We defie two (p + ) (p + ) matrices First, we defie the matrix R as follows: 0 0 0 λ p λ p λ p p+ λ p λ p λ p p+ R = λ λ λ p+ ad the diagoal matrix D as follows: λ D = λ p+ where the λ i s are the eigevalues of the matrix Q p for i p + We give the followig theorem for the computig the sums of the geeralized Fiboacci p- umbers from to by usig a matrix method Theorem 3 Let the sums of the geeralized Fiboacci umbers S be as i (7) The S = F p ( + p + ) Proof If we compute the det R by the Laplace expasio of determiat with respect to the first row, the we obtai that det R = det V, where the Vadermode matrix V is as i Theorem 3 Therefore, we ca easily fid the eigevalues of the matrix R Sice the characteristic equatio of the matrix R is ( x p x p ) (x ) ad by Lemma, the eigevalues of the matrix R are, λ,, λ p+ ad distict So the matrix R is diagoalizable We ca easily prove that T R = RD, where the matrices T, R ad D are as i (6), (9) ad (0), respectively The we have T R = RD Sice T = C, we write that C R = R D We kow that S = (C ), By a matrix multiplicatio, ( p ) S F p ( + i) = () i=0 By the defiitio of the geeralized Fiboacci p-umbers, we kow that p i=0 F p( + i) = F p ( + p + ) The we write the Eq () as follows: S F p ( + p + ) = (9) (0) ()
70 E Kilic / Europea Joural of Combiatorics 9 (008) 70 7 Thus, S = F p (i) = F p ( + p + ) i= So the proof is complete Fig I [30], the author presets a eumeratio problem for the paths from A to c, ad the shows that the umber of paths from A to c are equal to the th usual Fiboacci umber Now, we are iterested i a problem of paths The problem is as i Fig It is see that the umber of path from A to c, c, c p+ is Also, we kow that the iitial coditios of the geeralized Fiboacci p-umbers, that is, F p (), F p (),, F p (p + ), are Now we cosider the case > p + The umber of the path from A to c p+ is By the iductio method, oe ca see that the umber of the path from A to c is the th geeralized Fiboacci p-umber Refereces [] BA Bodareko, Geeralized Pascal s Triagles ad Pyramids: Their fractals, Graphs, ad Applicatios, Fiboacci Associatio, 993 [] WYC Che, JD Louck, The combiatorial power of the compaio matrix, Liear Algebra Appl 3 (996) 6 78 [3] MC Er, Sums of Fiboacci umbers by matrix methods, Fiboacci Quart (3) (984) 04 07 [4] S Falco, A Plaza, The k-fiboacci hyperbolic fuctios, Chaos Solitos Fractals, doi:006/chaos00609 [5] S Falco, A Plaza, The k-fiboacci sequece ad the Pascal -Triagle, Chaos Solitos Fractals 33 () (007) 38 49 [6] MJG Gazale, From Pharaos to Fractals, Priceto Uiversity Press, Priceto, New Jersey, 999 (Russia traslatio, 00) [7] VE Hoggat, Fiboacci ad Lucas Numbers, Houghto-Miffli, PaloAlto, Califoria, 969 [8] J Kappraff, Coectios The Geometric Bridge Betwee Art ad Sciece, secod ed, World Scietific, Sigapore, New Jersey, Lodo, Hog Kog, 00 [9] E Kilic, The geeralized order-k Fiboacci-Pell sequece by matrix methods, doi:006/cam006007 [0] E Kilic, D Tasci, The geeralized Biet formula, represetatio ad sums of the geeralized order-k Pell umbers, Taiwaese J Math 0 (6) (006) 66 670 [] E Kilic, D Tasci, O the geeralized order-k Fiboacci ad Lucas umbers, Rocky Moutai J Math 36 (6) (006) 95 96 [] C Levesque, O mth-order liear recurreces, Fiboacci Quart 3 (4) (985) 90 93 [3] B Rozi, The Golde Sectio: A morphological law of livig matter, Avaible from wwwgoldesectioet [4] NA Solaicheko, BN Rozi, Mystery of Golde Sectio, Theses of the Coferece Feid-90: No-traditioal ideas about Nature ad its pheomea, (3) Homel, 990 [5] CP Spears, M Bickell-Johso, Asymmetric Cell Disio: Biomial Idetities for Age Aalysis of Mortal vs Immortal Trees, i: Applicatios of Fiboacci Numbers, vol 7, 998, pp 377 39
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