A Combinatoric Proof and Generalization of Ferguson s Formula for k-generalized Fibonacci Numbers

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1 Jue 5 00 A Combiatoric Proof ad Geeralizatio of Ferguso s Formula for k-geeralized Fiboacci Numbers David Kessler 1 ad Jeremy Schiff 1 Departmet of Physics Departmet of Mathematics Bar-Ila Uiversity, Ramat Ga 5900, Israel kessler@dave.ph.biu.ac.il, schiff@math.biu.ac.il Various geeralizatios of the Fiboacci umbers have bee proposed, studied ad applied over the years (see [5] for a brief list). Probably the best kow are the k-geeralized Fiboacci umbers F (k) (also kow as the k-fold Fiboacci, k-th order Fiboacci, k-fiboacci or polyacci umbers), satisfyig F (k) = F (k) 1 + F (k) F (k) k, k, F (k) = 0, 0 k, F (k) k 1 = 1. A exhaustive bibliography of papers o the k-geeralized Fiboacci umbers would cover pages, so we just give a few refereces. The paper of Miles [9] seems to be the oldest wellkow paper o the subject, though Kuth [6] (sectio 5.4.) cites a work of Schlegel [13] datig from Numerous iterestig results ca be foud i the pages of the Fiboacci Quarterly, see for example [, 3, 7]. There are sigificat applicatios i computer sciece [6] ad probability theory [11, 4, 8], the latter of which will be importat for our purposes. Also much is kow about weighted k-geeralized Fiboacci umbers, with differet coefficiets i the recursio relatio, see for example [1]. I this paper we look at a particular case of weighted k-geeralized Fiboacci umbers, which we call the (k, p)-geeralized Fiboacci umbers. We defie these by = 1 p (k,p) (F k p ), k, (1) = 0, 0 k, () p k 1 = 1 p. (3) 1

2 Here k is a iteger ad p is a real with 0 < p < 1. For p = 1 we recover the k-geeralized Fiboacci umbers. The reader will have o trouble checkig that For fixed k, p we have a represetatio k+ = p, = 0, 1,...k 1. (4) = where λ 1,...,λ k are the roots (presumed distict) of k C i λ i, (5) λ k = 1 p p (λk 1 + λ k λ + 1), ad C 1,...,C k are chose to satisfy the iitial coditios ()-(3). The behavior of the λ i for geeral p ad sufficietly large k is similar to the behavior for p = 1 described i [9]: there is a real root, which we will call λ 1, betwee λ = 1 ad λ = 1, ad the remaiig p roots are all iside the circle λ = 1. The cotributio from λ 1 domiates the sum (5), ad ( ) (k,p) i particular lim F +1 /F (k,p) = λ1. As k icreases, λ 1 coverges rapidly to 1. p The focus of this paper will be o a iterestig formula for the F (k,p), very differet from the Biet-type represetatio (5). I the case of the stadard k-geeralized Fiboacci umbers (p = 1 ), ad i particular i the case of the stadard Fiboacci umbers (p = 1, k = ), our formula reduces to a expressio foud by Ferguso []. Ferguso proved his result usig geeratig fuctios; we offer a combiatoric proof, which highlights the importace of the the (k, p)-geeralized Fiboacci umbers i success ru problems. The combiatoric proof icely explais why F (k,p) depeds o the greatest iteger ot exceedig ( k + 1)/(k + 1). This iteger is simply the maximum possible umber of distict rus of k successes i a sequece of k trials. Notatio. Let H(x) deote the Heaviside fuctio defied by { 1 x 0 H(x) = 0 x < 0 Theorem. For k = k+1 k+1 ( ) ( )] k(r + 1) k(r + 1) ( 1) r p k(r+1) (1 p) [(1 r 1 p) +. (6) r r 1 Here we uderstad that for ay o-egative iteger N, x deotes the largest iteger ot exceedig x. ( N 1 ) = 0, ad for ay real x,

3 Proof. Let P (k,p) deote the probability that there is at least oe ru of at least k successes i a sequece of idetical Beroulli trials, each havig probability p of success. The possible outcomes of the trials ca be partitioed as follows: either the first trial results i failure, or the first trial results i success ad the secod i failure, or the first two i success ad the third i failure,..., or the first k 1 result i success ad the kth i failure, or the first k all result i success. Thus, usig the law of total probability we have = q( 1 + p + p p k 1 k ) + pk, k, (7) where we have writte q = 1 p. The iitial coditios for this recursio are = 0, 0 k 1. If we write R (k,p) with iitial coditios R (k,p) that R (k,p) = k+, or = 1 p R (k,p) we fid = q p (R(k,p) 1 + R (k,p) R (k,p) k ), k, = p, 0 k 1. Comparig with (1) ad (4) we deduce = p k (1 k ), k. (8) We ow cosider computig usig the iclusio exclusio priciple. Let A i, i = 1,,..., k+1, deote the evet that there is a miimal ru of k successes startig o the ith trial. By miimal we mea that there is a ru of precisely k successes startig o the ith trial. Thus for i = 1,,..., k, beig i A i meas that trials i, i+1,...,i+k 1 result i success, but trial i + k results i failure (so i particular, the evet that the first k + 1 trials result i success but trial k + results i failure is cotaied i A but ot A 1 ). For i = k +1, beig i A i meas just that the last k trials (trials k +1, k +,..., ) result i success. With this defiitio of the evets A i, P (k,p) is simply the probability of their uio. So applyig the iclusio exclusio priciple we have = P(A i ) P(A i A j ) + P(A i A j A l ) i i<j i<j<l ( 1) r 1 P(A i1 A i... A ir ) i 1 <i <...<i r Computig the first term is straightforward. We have P(A i ) = { p k q 1 i k p k i = k + 1, ad so P(A i ) = ( p k q( k) + p k) H( k). (9) i 3

4 The factor H( k) here reflects the fact that uless k we caot have a success ru. Movig to the secod term, we ca oly have a pair of success rus if k + 1. Because of our choice of A i as the evet that there is a miimal success ru startig at the ith trial, A i ad A j are mutually exclusive if j i k, ad i full geerality we have Thus 0 j i P(A i A j ) = (p k q) 1 i, j > i + k, j < k + 1 p k q 1 i, j > i + k, j = k + 1. P(A i A j ) = i<j = k 1 k j=i+k+1 p k q + k p k q ( 1 pk q ( k)( k 1) + p k q( k)) H( k 1). The third term i the iclusio exclusio priciple is P(A i A j A l ) = i<j<l = 3k k 1 k j=i+k+1 l=j+k+1 p 3k q 3 + 3k 1 k j=i+k+1 p 3k q ( 1 6 p3k q 3 ( 3k)( 3k 1)( 3k ) + 1 ) p3k q ( 3k)( 3k 1) H( 3k ). A patter is clearly emergig. For arbitrary r 1 we have i 1 <i <...<i r P(A i1 A i... A ir ) = + +1 r(k+1) i 1 =1 + r(k+1) i 1 =1 +1 (r 1)(k+1) i =i 1 +(k+1) + (r 1)(k+1) i =i 1 +(k+1) (+1) (k+1) i r=i r 1 +(k+1) + (k+1) i r 1 =i r +(k+1) p rk q r p rk q r 1. Lemma. For positive itegers N, r, K with N > rk N rk i 1 =1 N (r 1)K i =i 1 +K... N K i r=i r 1 +K 1 = ( ) N rk + r 1 r. (10) Proof. We cosider arragemets of N 1 objects with the costraits that there are K objects of type 1 that appear i successio i a prescribed order, followed (ot ecessarily immediately) by K objects of type that appear i successio i a prescribed order, followed i tur by K objects of type 3 that appear i successio i a prescribed order, ad so o, up to K objects of type r (that appear i successio i a prescribed order). Furthermore, the 4

5 remaiig N 1 Kr objects are idetical. The left had side of (10) couts the umber of such arragemets by coutig the ways to place the first object of type 1, the first object of type etc. A more sesible way to cout, however, is to otice that we ca treat each of the blocks of objects of types 1,,..., r as metaobjects, ad the we are coutig arragemets of just N 1 r(k 1) objects i which r have prescribed order ad the remaiig N 1 Kr are idetical. The umber of such arragemets is clearly just the umber of ways to chose r from N rk + r 1. Returig ow to the proof of the mai theorem, usig the lemma we ca write dow all the terms i the iclusio exclusio priciple, ad we have = [( ) ( ) ] rk rk ( 1) r 1 p rk q r + p rk q r 1 H (( + 1) r(k + 1)). r=1 r r 1 The Heaviside fuctios restrict the sum to be over a fiite rage, ad thus we reach the fial result = +1 k+1 r=1 ( ) ( )] rk rk ( 1) r 1 p rk (1 p) [(1 r 1 p) + r r 1. (11) Usig this i (8) we obtai the result i the theorem. Corollary 1.[] For k F (k) = k+1 k+1 [ ( ) ( )] 1 k(r + 1) k(r + 1) ( 1) r +1 kr k r + r r 1. (1) Corollary.[] For F = 1 3 [ ( ) ( )] 1 r r ( 1) r 3r 1 + r r 1. (13) Corollary 3. Defie Σ (k,p) N = k+1 N ( ) ( ) N kr N kr r ( ) r 1 p 1. r p p (This is the sum of terms i a diagoal i a geeralizatio of Pascal s triagle, see [1].) The = p ( Σ (k,p) k+1 Σ(k,p) k). (14) (The proof of this is a elemetary maipulatio of biomial coefficiets.) 5

6 Commet 1. It is well-kow that F = 1 ( ) 1 r r, i.e. that the stadard Fiboacci umbers ca be expressed as a diagoal sum of Pascal s triagle. I (14) we see that for ay k, the k-geeralized Fiboacci umbers ca be expressed as the differece of two diagoal sums of a geeralized Pascal s triagle. I [1] it is show that the Σ (k,p) obey the recursio relatio Σ (k,p) = 1 p Σ(k,p) 1 + p 1 p Σ(k,p) k 1. The roots of the characteristic polyomial of this recursio are exactly those of the recursio (1), plus the root 1. Thus we should be able to write the F (k,p) the Σ (k,p). The simplicity of (14), however, is remarkable. as a liear combiatio of Commet. There is a iterestig way to geerate the result (11) from the recursio relatio (7). We write (7) i the form = q k p i 1 i + p k, k, = 0, 0 < k. (15) Cosider the modified recursio = q p i 1 i + p k, k, = 0, < k, (16) which determies quatities P (k,p), where ow rus over all the itegers, eve egative. We ca regard the exact solutio as a first approximatio to the P (k,p). Remarkably (16) has the = p k (1 + q( k))h( k), as ca be verified by a tedious calculatio. This is precisely the first term (9) i the iclusio exclusio priciple! We cotiue by writig a recursio for the differeces Q (k,p) P (k,p). Subtractig (16) from (15) gives = Q (k,p) k = q p i 1 Q (k,p) i q p i 1 i i=k+1 k = q p i 1 Q (k,p) k i q p i+k 1 (1 + q( i k)) i=k+1 k = q p i 1 Q (k,p) i q( k)p k H( k 1), k. (17) 6

7 This recursio should be solved with iitial coditios Q (k,p) = 0, 0 < k, ad i fact we clearly have Q (k,p) = 0 for 0 < k +1. Agai, we approximate (17) with the recursio Q (k,p) = q with iitial coditios Q (k,p) give p i 1 Q (k,p) i q( k)p k, k + 1, (18) = 0, < k + 1. This recursio ca also be solved exactly, to Q (k,p) = p k q( k) (1 + 1 ) q( k 1) H( k 1), which is the secod term i the iclusio exclusio formula (with the correct sig). This procedure ca be cotiued. There may be some relatio with -geeralized Fiboacci umbers [10]. Ackowledgmets D.K. ad J.S. both ackowledge support of the Israel Natioal Sciece Foudatio. Refereces [1] F.Dubeau, W.Motta, M.Rachidi ad O.Saeki, O weighted r-geeralized Fiboacci sequeces, Fiboacci Quart. 35 (1997) [] D.E.Ferguso, A expressio for geeralized Fiboacci umbers, Fiboacci Quart. 4 (1966) [3] H.Gabai, Geeralized Fiboacci k-sequeces, Fiboacci Quart. 8 (1970) [4] A.P.Godbole, Specific formulae for some success ru distributios, Statist. Prob. Lett. 10 (1990) [5] V.C.Harris ad C.C.Styles, A geeralizatio of Fiboacci umbers, Fiboaci Quart. (1964) [6] D.Kuth, The art of computer programmig. Volume 3. Sortig ad Searchig. Addisio Wesley (1973). [7] G.-Y.Lee, S.-G.Lee, J.-S.Kim ad H.-K.Shi, The Biet formula ad represetatios of k-geeralized Fiboacci umbers, Fiboacci Quart. 39 (001)

8 [8] F.-H.Li, W.Kuo ad F.Hwag, Structure importace of cosecutive-k-out-of- systems, Op.Res.Lett. 5 (1999) [9] E.P.Miles, Jr., Geeralized Fiboacci umbers ad associated matrices, Amer. Math. Mothly 67 (1960) [10] W.Motta, M.Rachidi ad O.Saeki, O -geeralized Fiboacci sequeces, Fiboacci Quart. 37 (1999) 3-3. [11] A.N.Philippou ad A.A.Muwafi, Waitig for the Kth cosecutive success ad the Fiboacci sequece of order K, Fiboacci Quart. 0 (198) 8-3. [1] J.A.Raab, A geeralizatio of the coectio betwee the Fiboacci sequece ad Pascal s triagle, Fiboacci Quart. (1963) [13] V.Schlegel, El Progresso Matematico 4 (1894) MSC000 Classificatio Numbers: 11B39, 60C05. 8