The r-generalized Fibonacci Numbers and Polynomial Coefficients

Transcription

1 It. J. Cotemp. Math. Scieces, Vol. 3, 2008, o. 24, The r-geeralized Fiboacci Numbers ad Polyomial Coefficiets Matthias Schork Camillo-Sitte-Weg Frakfurt, Germay Abstract A simple expressio for the r-geeralized Fiboacci umbers is give i terms of polyomial coefficiets, geeralizig i a straightforward way the well-kow expressio of the Fiboacci umbers i terms of biomial coefficiets. Keywords: Fiboacci umbers, polyomial coefficiets 1 Itroductio The r-geeralized Fiboacci umbers F (r) recursio relatio F (r) (with r 2) are defied by the = F (r) 1 + F (r) F (r) r (1) for r together with the iitial values F (r) i = δ i,r 1 for 0 i r 1 [11]. Clearly, the case r = 2 correspods to the covetioal Fiboacci umbers, i.e., F (2) F, ad the case r = 3 correspods to the Triboacci umbers. ca be writte i the form Miles showed i [11] that the F (r) F (r) 0 k 1,..., kr k 1 +2k 2 + +rkr= r+1 (k k r )! k 1! k r! which he cosidered as the geeralizatio of the followig idetity for r = 2 F (2) F 1 2 ( ) 1 m m (where we have deoted by x the greatest iteger less tha or equal to x). The explicit form of the r-geeralized Fiboacci umbers (or the weighted aalogue) has bee discussed from may differet poits of view by may authors, (2) (3)

2 1158 M. Schork see, e.g., [3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15]. Let us ote that (3) shows that the Fiboacci umbers ca be expressed through the left-justified biomial coefficiets of Hoggatt ad Bickell [6]. The same authors derived i [6] a expressio for F (r) i terms of the left-justified r-omial coefficiets. I the preset ote we will derive aother (straightforward) geeralizatio of (3) for the F (r) (ad their weighted aalogue) by usig the polyomial coefficiets istead of the multiomial coefficiets used i most of the refereces give above. Sice the polyomial coefficiets are certai sums of multiomial coefficiets there exists, of course, a very close coectio to the results obtaied i the refereces give above. Before startig let us metio that the r-geeralized Fiboacci umbers are begiig to play a role i physics, e.g., i the cotext of geeralized exclusio statistics (which geeralize the well-kow Bose-Eistei statistics for bosos ad the Fermi-Dirac statistics for fermios) [16, 17] ad geeralized Heiseberg algebras [18]. 2 The polyomial coefficiets Followig Comtet [2] we itroduce the polyomial coefficiets,k (deoted by ( ),r+1 k i [2]) by the idetity (1 + x + + x r ) r k=0,k xk. (4) The case r = 1 leads via the biomial formula directly to the biomial coefficiet A (1),k = ( ) k. I the geeral case oe may use the multiomial formula for the left-had side of (4) to obtai,k = 0 k 1,...,kr k 1 +2k 2 + +rkr=k! ( k 1 k r )!k 1! k r!. (5) Specializig (5) to the case r = 2 oe obtais for 0 k the followig explicit expressio for the triomial coefficiets k A (2) 2 k,k = 2! ( )( ) ( k + l)!(k 2l)!l! = k l. (6) k l l Let us itroduce the Gegebauer polyomials C(t) ν through their geeratig fuctio [1], (1 2tx + x 2 ) ν = k=0 Ck ν(t)xk. Choosig t = 1, ν = ad 2 comparig this to (4) shows that A (2),k = C k ( 1 ). (7) 2

3 Geeralized Fiboacci umbers ad polyomial coefficiets Fiboacci umbers ad polyomial coefficiets As a first step, let us rewrite (2) by collectig those tuples (k 1,...,k r ) with k 1 + +k r = m. This implies k 1 +2k 2 + +rk r = m+k 2 +2k 3 + +(r 1)k r, so that m ca be at most r + 1 (ad this occurs for (k 1 = m, 0,...,0)). Therefore, F (r) r+1 = k 1 +k 2 + +kr=m k 1 +2k 2 + +rkr= r+1 m! k 1!k 2! k r 1!k r!. Comparig this to (5) shows that the (r + 1)-geeralized Fiboacci umbers are give by F (r+1) r m, r m. (8) To brig this ito a more coveiet form like (3), we first switch to a ew idex m l r m. This yields o the right-had side r A(r) r l,l. However, the coefficiets,k are ovaishig oly if 0 k r. This implies i our case that l r( r l), or, equivaletly, l r( r). Reamig r+1 the idex l m ad usig Comtet s otatio for the polyomial coefficiets,,k ( ),r+1 k, we have fially show the followig theorem. Theorem 3.1 The r-geeralized Fiboacci umbers ca be expressed through the polyomial coefficiets as F (r) (r 1)( (r 1)) r ( ) (r 1) m, r. (9) m Example 3.2 For r =2formula (9) reduces to (3) sice ( ) ( ),2 k k. Example 3.3 For r =3we ca combie (9) with the explicit formula (6) for the triomial coefficiets to obtai the followig explicit formula for the Triboacci umbers i terms of biomial coefficiets, F (3) 2( 2) 3 m 2 ( )( ) 2 m m l. (10) m l l Recallig the coectio (7) of the triomial coefficiets to the Gegebauer polyomials, we ca alteratively write F (3) 2( 2) 3 C +m+2 m ( 1 2 ).

4 1160 M. Schork 4 Geeratig fuctios ad the geeral case Istead of the direct approach from above oe ca also use geeratig fuctios as follows. The recursio relatio ad the iitial values imply that the geeratig fuctio for the r-geeralized Fiboacci umbers is give by F (r) (z) F (r) z 0 z r 1 1 z z 2 z r. (11) Itroducig f r (z) :=z +z 2 + +z r, oe obtais F (r) (z) =z r 1 m 0[f r (z)] m. However, usig [f r (z)] m = z m (1 + z + + z r 1 ) m = z m this immediately implies F (r) (z) =z r 1 m 0 A (r 1) m,l z l (12) A (r 1) m,l z m+l. (13) F (r) is give as the coefficiet of z, i.e., m + l + r 1. I particular, this implies that m r + 1 as well as l = r +1 m, so that (r 1) F (r) = A (r 1) m, (r 1) m. (14) This is exactly (8) up to a shift r +1 r. Note that expressig [f r (z)] m ot as i (12) but writig (1 + + z r 1 ) m = ( ) 1 z r m 1 z =(1 z r ) m (1 z) m ad usig the biomial formula ad the geometric series, it is possible to express F (r) through a sum of products of biomial coefficiets aalogous to (10), see [9]. The above discussio ca be easily geeralized to the weighted case. If a =(a 0,a 1,...,a r 1 ) C r is a vector of weights, the weighted r-geeralized Fiboacci umbers are defied by the recursio relatio = a a a r 1 r (15) together with the iitial values i for i such that 0 i r 1. The geeratig fuctio of the F (r),a is give by [10] (z) F (r),a z v 0 + v 1 z + + v r 1 z r a 0 z a 1 z 2 a r 1 z, r where the v i (for 0 i r 1) are defied through the iitial values by v i = i j=0 i j a j 1 (16)

5 Geeralized Fiboacci umbers ad polyomial coefficiets 1161 (with the covetio a 1 = 1), ad geeralizes (11). Proceedig as above ad defiig the weighted polyomial coefficiets A (r 1),a,k ( ),r a k i aalogy to (4) by [a 0 + a 1 z + + a r 1 z r 1 ] m = A (r 1),a m,l z l oe directly obtais the geeralizatio of (13), i.e., (z) ={v 0 + v 1 z + + v r 1 z r 1 } m 0 Therefore, oe has r 1 F (r),a = i i=0 ( ) m, r a z l, l A (r 1),a m,l z m+l. v i A (r 1),a m, i m. (17) This is essetially the result show by Levesque i [10] i a slightly differet form (see also the related formulas derived i [6, 9, 13]). We wat to brig this ito the form used i (9). For this we agai switch the idex m l i m ad observe that this yields for the ier sum i v ia (r 1),a i l,l. Usig that the summads are ovaishig oly if l (r 1)( i l), or l (r 1)( i), r we have foud after the relabellig l m that r 1 (r 1)( i) r i=0 v i A (r 1),a i m,m. Fially, switchig to the aalogue of Comtet s otatio, A (r 1),a k,l ca summarize the result i the followig theorem. ( ) k,r a l,we Theorem 4.1 The weighted Fiboacci umbers defied through (15) ad the iitial values i (for i such that 0 i r 1) are give for r by r 1 (r 1)( i) r i=0 where the coefficiets v i are defied by (16). ( ) i m, r a v i, (18) m Example 4.2 Let us cosider a = 1 =(1, 1,...,1), i.e., the uweighted case. Assumig the iitial values i = δ i,r 1 (i.e., oly the coefficiet v r 1 = 1 is ovaishig), this is exactly the situatio cosidered i the last sectio. Clearly, due to v i = δ i,r 1 ad ( ) ( ) k,r 1 l k,r l equatio (18) reduces to (9).

6 1162 M. Schork Refereces [1] M. Abramowitz ad I.A. Stegu, Hadbook of Mathematical Fuctios, (Natioal Bureau of Stadards, 1964). [2] L. Comtet, Advaced Combiatorics, Reidel, Dordrecht, [3] F. Dubeau, O r-geeralized Fiboacci umbers, Fiboacci Q. 27 (1989), [4] F. Dubeau, W. Motta, M. Rachidi ad O. Saeki, O weighted r-geeralized Fiboacci sequeces, Fiboacci Q. 35 (1997), [5] L.E. Fuller, Solutios for geeral recurrece relatios, Fiboacci Q. 19 (1981), [6] V.E. Hoggatt ad M. Bickell, Geeralized Fiboacci polyomials, Fiboacci Q. 11 (1973), [7] G.-Y. Lee ad S.-G. Lee, A ote o geeralized Fiboacci umbers, Fiboacci Q. 33 (1995), [8] G.-Y. Lee, S.-G. Lee, ad H.-G. Shi, O the k-geeralized Fiboacci matrix Q k, Liear Algebra Appl. 251 (1997), [9] G.-Y. Lee, S.-G. Lee, J.-S. Kim ad H.-K. Shi, The Biet formula ad represetatios of k-geeralized Fiboacci umbers, Fiboacci Q. 39 (2001), [10] C. Levesque, O m-th order liear recurreces, Fiboacci Q. 23 (1985), [11] E.P. Miles, Geeralized Fiboacci umbers ad associated matrices, Amer. Math. Mothly 67 (1960), [12] M. Moulie ad M. Rachidi, Suites de Fiboacci r-gééralisées, théorème de Cayley-Hamilto et chaîes de Markov, Red. Semi. Mat. Messia, Ser. II 19 (1997), [13] M. Moulie ad M. Rachidi, Applicatio of Markov chais properties to r-geeralized Fiboacci sequeces, Fiboacci Q. 37 (1999), [14] A.N. Philippou, A ote o the Fiboacci sequece of order k ad the multiomial coefficiets, Fiboacci Q. 20 (1983), [15] A.N. Philippou, C. Georghiou ad G.N. Philippou, Fiboacci polyomials of order k, multiomial expasios ad probability, Iterat. J. Math. Math. Sci. 6 (1983),

7 Geeralized Fiboacci umbers ad polyomial coefficiets 1163 [16] M. Rachidi, E.H. Saidi ad J. Zerouaoui, Fractioal statistics i terms of the r-geeralized Fiboacci sequeces, It. J. Mod. Phys. A 18 (2003), [17] M. Schork, Some algebraical, combiatorial ad aalytical properties of paragrassma variables, It. J. Mod. Phys. A 20 (2005), [18] M. Schork, Geeralized Heiseberg algebras ad k-geeralized Fiboacci umbers, J. Phys. A: Math. Theor. 40 (2007), Received: February 13, 2008